For about a hundred of our Earth years, our planet has been a noisy little mudball when it comes to radio signals. A very common plot amongst science fiction authors is to depict aliens as having made contact with Earth culture via stray TV broadcasts.
One bit of science that these writers surprisingly get right consistently is that radio signals propagate at light speed. Given that on TV distances are conveniently measured in light years, it's an easy conversion formula: aliens 50 light-years from Earth are just now getting TV signals sent in the 1960's, thus the visitor that shows up, having skipped the intervening distance via Faster-than-Light Travel, will talk and dress like a beatnik in an attempt to fit in. Hilarity Ensues, or it provides a vital clue to the protagonist that something isn't quite right about this guy.
There are some major technical problems with this concept, but most writers will ignore them. First of all, non-directional broadcast signals (like TV and radio signals) can not be received beyond a fraction of a light year even by much more powerful telescopes than the ones available , so it shouldn't be that surprising that we have not been receiving any of their transmissions, even if they have radio technology (or maybe they all read books on their planet). Directed radio signals (like radar signals) can be possibly received thousands of light years away but can be detected in a far smaller area and may not even be recognized as sign of extra-terrestrial life. Another scientific theory is that they've developed a different form of communication that doesn't depend on radio broadcasts, and all of their surviving transmissions from when they did have already passed us by.
Regardless, misunderstandings and misinterpretations about Earth culture and human behavior from tiny snippets of old sitcoms are comedy gold especially if it means they expect Earth to be like that, if they learn their English from it it justifies Aliens Speaking English and so the concept keeps coming back up. Compare Alien Arts Are Appreciated. See also Do Not Adjust Your Set for when the aliens are broadcasting to us instead.
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Anime and Manga
A major plot point in Super Dimension Fortress Macross, starting with the Miss Macross pageant and culminating in the battle against the Bodolza fleet. It neatly sidesteps the distance and signal strength issues by having the transmission come from the titular ship, and the Zentradi warships chasing it are (in astronomical distances) a stone's throw away.
In the Sentai anime Shinesman, the ditzy female villain once asks about the heroes, but she's got the completely wrong idea about them. A nearby mook admits, when a smarter villain asks, that they didn't have enough footage of the actual heroes so they had to fill in the blanks in their report with examples from a sentai tv-series.
The Keronians from Keroro Gunsou seem to be well-acquainted with Earth culture. In an early chapter of the manga, Natsumi is surprised when Keroro takes offense to being compared to Q-taro the Ghost (Slimer in the Tokyopop translation): "How dare you compare me to that overeating ectoplasmic idiot?!"
Evidently, Pekopon (Earth) is an entertainment Mecca and produces most of the galaxy's highest rated TV shows.
In Haiyore! Nyarko-san, Earth's entertainment is popular amongst aliens. So popular in fact, that there's even a Black Market for manga, anime, C Ds, DVDs, Blu-Ray, comic-books, movies, etc. And apparently, this is a large enough problem that it warrants sending border patrol/contraband agents planetside to make sure nothing makes it off the planet illegally (though Nyarko-san has no qualms about sneaking her own private collection of purchased media off the planet; after all, "Its not a crime if you don't get caught"). Our entertainment industry is considered our primary resource, and our major appeal on the galactic stage.
Book four of 2000AD's Nemesis the Warlock is set in the Gothic Empire, populated by a race of shapeshifting aliens who received the first large scale radio transmissions of the 1920's. They promptly based their society on what they thought was Earth's pre-1914 Golden Age, particularly on Victorian society and the British Empire (even with their own version of Jack the Ripper).
This trope is a part of the origin of The Savage Dragon. He was a warlord who wanted to subjugate Earth despite the fact that his race is pretty peaceful. They decide to rip out chunks of his brain in order to give him amnesia (he has a healing factor). Since they were monitoring Earth for years, they used their satellite feeds to give Dragon new memories and dumped him in Chicago.
In the X-Men books, the residents of the Mojoverse have had their dreams bombarded with TV transmissions from Earth for thousands of their years (time works differently in their dimension). Sort of like the Star Trek Iotians, this exposure turned them into a Dimension of Hats organized around emulation of television, to the point that their Dimension Lord is whichever network executive has the best ratings.
Referenced in Atomic Robo by Carl Sagan, during an "unmanned" mission to mars.
Sagan: Gentlemen, we're communicating across the vast, empty chasm of space. Do you really want our first interaction with another intelligent species to be the words "robot pornography?"
After the Masquerade gets smashed to bits in Journey, the TSAB enters into more open contact with Earth and develops a liking for Terran culture and entertainment. There is a passing mention of a Top Gear special on Mid-Childa. Complete with the Stig's Magical Girl cousin.
The Thermians in Galaxy Quest take the fictional events of old television shows seriously, calling them "historical documents." Besides thinking that the main characters really are space explorers as opposed to the actors who played them, they weep for "those poor people" stranded on Gilligan's Island. It is explained that the Thermians have no concept of acting, or even pretending because they were only recently introduced to the concept of deception.
In Contact, the extraterrestrials in the Vega system receive the first transmission strong enough to reach outer space - a transmission of Adolf Hitler's opening speech at the 1936 Olympic Games; the aliens send the signal back to Earth, combined with a sequence of prime numbers and blueprints for a machine.
The scientists of the film also point out that the aliens could not possibly have understood the historical context of the transmission (or even what was being said) and state that their transmitting it back would simply have been to show that it had been received (as well as to carry the blueprints). They mock the idea held by the government and military officials that it must mean they've made contact with alien Nazis.
David Drumlin: ...'36 Olympics was the first television transmission of any power that went in to space. That they recorded it, and sent it back, is simply a way of saying "hello, we heard you."
Michael Kitz: Or, "Sieg Heil, you're our kind of people."
Of course, in the original book, we learn that the aliens specifically chose to beam back that transmission because they did understand the cultural context and wanted to warn us against following that particular route.
The Contact scenario also avoids the difficultly of decoding it, as they could have just been retransmitting it without knowing even what kind of signal it was.
We know that they must have decoded it, because they didn't just send it back unaltered. It was sent back with the original frames alternating with the data frames containing the design data for The Machine. This was likely the first signal they were able to pick up that was high enough bandwidth for this to be feasible.
The Three Stooges in Orbit. The Martian videophone accidentally starts broadcasting Earth television, rapidly convincing the Martian Big Bad that rather than conquer the Earth, it would be better to wipe it from existence.
Variant: In the first Transformers live-action film, Optimus Prime and the Autobots learned English from the Internet. Thus, you have Optimus Prime saying "My bad".
In the novelization of the movie, Optimus first tries to address Sam in Mandarin. The explanation given is that this is the language spoken on Earth by the most humans, completely leaving aside the fact that most international communication does in fact take place in English, to say nothing that the most cursory scan of the surrounding transmission waves would make it obvious that they were in an English-speaking locale.
The Rifftrax of the film has a little fun with this concept. Optimus demands to be taken to "KingSchnappi".
In Highlander II The Quickening it can inferred this is the case from the alien General Katana's fondness for pop culture references. Apparently there isn't a lot to do on Zeist.
Dave Barry makes fun of this in several in his columns. The best example is when he theorizes that aliens developed a fondness for bad TV commercials and threatened the government to keep playing them, which is why so many bad commercials are on air.
In the novel version of 2001: A Space Odyssey, the Sufficiently Advanced Aliens that are going to turn Dave into the Star Child first calm him by giving him a mock-up of a hotel suite. They don't get all the details right, though: he's disturbed when he notices drawers won't open and the books are part of the bookcase. What happened becomes clear when he turns on the TV and sees a scene from a movie set in a hotel suite exactly like the one he is in.
Animorphs used a version of this, albeit without the "long distance" part (but, interestingly, still using the trope as an Homage to 1980s TV): the body-snatching aliens first visit Earth in 1991, and panic upon discovering, while in orbit, news reports indicating that humans have mastered Faster-than-Light Travel and Energy Weapons. They quickly realize that it's not real, and conclude that human indulgence in escapism makes us an even better target.
They decide to land in Hollywood, (instead of New York, Washington Dee Cee, or Ellay) because it is obviously the most important due to the amount of times it is mentioned in the decidedly factual parts of television broadcasts.
Also contains a quite literal version- Aximilli steals cable, and records everything for later reference. His excuse is to screen for Yeerk propaganda (which actually works several times), more often than not, he watches soap operas and 'These Messages' which he finds more amusing than most other shows.
Not to mention that Andalite kids apparently watched (or rather, pretended to watch while basically doing anything but) human television. They watched the news, entertainment, and...
Tobias: Music? You mean like MTV? You were watching music videos on the Andalite home world?
In the novel Lacuna some Toralii understand English before encountering the Humans (and, given the book's content, probably Mandarin as well). How they learned the language is a bit of a mystery but it's probably this.
In the Doctor WhoPast Doctor Adventures novel Synthespians™, human colonists in the future do this with broadcasts from Earth. It's pointed out that until they had the help of the Nestene Consciousness, the shows were so degraded it was like watching it through a snow storm.
Phule's Company: In the novel No Phule Like an Old Phule, the Zenobians revere a figure called L'Vis which is actually from an old broadcast of Elvis.
Adrift Among The Ghosts by Jack L. Chalker, in which an alien race sentenced one of their criminals to criss-cross space at just the right distance from Earth to intercept and record historic radio and TV broadcasts. Why was this considered a punishment? Because it forced him to relive our nuclear holocaust over and over and over and over...
In Mostly Harmless, the Grebulons, a group of aliens who have lost their memories, fill their cultural vacuum by watching TV from their base on the 10th planet of our solar system.
Parodied in "The Holy Stomper vs. the Alien Barrel of Death". The aliens who pick up human soap operas and professional wrestling are a lost cult on a generation ship, and they base their entire new culture on the shows—but have no idea what they mean. Factor in their inhuman appearance, and their own broadcasts become utterly bizarre.
Spoofed in Star Smashers of the Galaxy Rangers by Harry Harrison. Every alien race the heroes come across has "listened to your radio broadcasts" and learnt fluent English for one reason or another.
The Antheans in The Man Who Fell to Earth observe Earth this way; in the movie adaptation, protagonist Thomas Jerome Newton, once he's established himself there in the guise of a human businessman, appears in a television ad for his Mega Corp that reaches his family as a greeting of sorts. At the end, when he is unable to return home, he records an album of music that includes a goodbye to them and his people, hoping they will hear it via radio waves...if they aren't already dead.
The alien invaders of The Killing Star ultimately decide that humankind is a threat because they intercepted old episodes of Star Trek sent out centuries ago and concluded that all of the Rubber Forehead Aliens were an indication that humans were so xenophobic that they would never accept relations with an alien species that was not humanoid.
Invoked in a Wishbone Mysteries book involving a UFO sighting in Oakdale. Trying to unmask a hoaxer pretending to be an alien over IRC, David asks him what his favorite human TV show was in an attempt to catch him violating the speed of light. The hoaxer doesn't fall for it, responding with I Love Lucy (appropriate given his claim that his homeworld was 40 light-years from Earth).
Live Action TV
The Adventures of Pete & Pete: Big Pete befriends a boy who dresses like he's from the 1950's, and who is obsessed with Johnny Unitas and the 1958 NFL Championship Game, which is credited with putting the NFL in the public consciousness and essentially making pro football "big", which featured Unitas leading the Colts to a 23-17 overtime victory over the New York Giants.
Star Trek: The Original Series' Trelane, the eponymous "Squire of Gothos", wasn't receiving radio signals, but clearly was limited by speed-of-light transmission when he thought that 18th-century fashions and behavior were the latest things for Earth people, there on his planet some 600 light years from Earth. Then again, he was merely a child from a race of Sufficiently Advanced Aliens and might be excused from making such a mistake.
In another episode ("A Piece of the Action"), it was discovered that the people of Iotia had based their entire culture on a book left behind by an earlier survey ship: "Chicago Mobs of the 20s." Hilarity Ensues when Kirk, Spock, Bones and even Scotty have to deal with cliche gangsters, curious local customs and slang, and the enigma of manual transmission.
The Star Trek: The Next Generation episode "First Contact" had an interesting reversal of this trope; here, the Federation monitors an alien civilization who are about to become capable of interstellar travel, and when introducing one of their head scientists to the greater galaxy notes that, among other things, they've been looking at their radio transmissions to learn more about them, causing her to comment, "I hope you don't judge us by our popular entertainment!"
Also Star Trek: The Next Generation: In "The Royale", the crew finds a planet with weird simulation of a cliche '30s-style gambling casino. As it turns out, aliens had accidentally made a few humans crash many years ago and tried to construct the only survivor a surrogate home based on the novel he had with him. Bad luck for him - He hated the book.
In the Tales From The Darkside episode "Distant Signals," a mysterious, eccentric investor brings together the cast and crew of a 20-year-old private eye TV series, which was cancelled before it got a proper ending, so the story can finally be resolved. The investor turns out to be the representative of an alien race who had been following the show. (In the original short story, the private eye show was a western.)
Similar to the Transformers example above, the aliens in The Greatest American Hero spoke to the main characters through piecing together radio signals in their car. Also, in the episode "Operation Spoilsport," the aliens repeatedly played the song "Eve of Destruction" to indicate to the titular hero that a nuclear war was about to start.
The Strangerers, a comedy serial by Rob Grant (one half of the Grant/Naylor partnership that created Red Dwarf) takes its concept directly from this trope - the aliens assume a 1950s identity, and tumble into all manner of jolly japes as they wrack their brains to remember this strange human practice of 'walking' and the lift for their hotel bedroom.
In Lexx, aliens are tipped off to the existence of life on Earth by Marconi's experiments with transatlantic radio.
While no aliens have been explicitly said to have discovered Earth because of our TV and radio, they certainly do enjoy it. Doctor Who's nemesis The Master is particularly pleased by our invention of the Teletubbies.
He was also somewhat enamored with The Clangers some decades previous.
The Filk song Extra-Terrestrial Outrage by Diana Gallagher has aliens arrive in the late 21st Century to declare war on Earth ... because they're angry about the cancellation of Star Trek: The Original Series.
The Split Enz song Poor Boy frames this trope in a romantic context.
Ziggy at least twice had aliens show up referencing Star Trek: one set who, like the Thermians from Galaxy Quest, didn't realize it was fiction and wanted to join the Federation ("Take us to Captain Kirk!"), and a couple of others who decided to complain to Ziggy about how silly the plots were getting "lately."
In the "Scared Spirits" supplemental adventure for the Ghostbusters Frightfully Cheerful RPG, the PKE-stealing aliens learned English by watching television broadcasts and speak "Madison Avenue-ese." That is to say, they are constantly spewing slogans from TV commercials. For some reason, any human they "zombify" by draining them of PKE also talks like this.
The Scrin frequently gather data from human networks in Command And Conquer Tiberium Wars. At the beginning of the Scrin campaign, the mothership's AI taps into satellite communications to run a cryptanalysis on the television broadcasts that appear throughout the other campaigns, learning Machine Monotone English from them. In an interesting subversion of this trope, the Scrin expect from experience that Tiberium drove humanity to near-extinction but after almost getting blown out of the sky by GDI ion cannons, they tap into military frequencies and eventually conclude that humanity is fragmented yes, but they're also "warlike to the extreme". Finally, the Supervisor intercepts the cutscene where Kane explains the part of his plan involving the Liquid Tiberium Bomb and realizes they've been duped into invading at least a century earlier.
Inverted in Commander Keen 4: Billy's radio receives signals from many light years away and consequently picks up a lot of alien sitcoms.
In Mass Effect 2, Legion mentions that the geth regularly mine the extranet for information on organics, seeking to better understand them.
Not only that, but they also perform social experiments by implanting false information to see what will happen, such as one time when they intentionally falsified a report claiming a star cluster resembled a Salarian goddess, that lead to several parties trying to lay claim to it before discovering it didn't actually exist. If they didn't lack emotions, you could almost suspect the Geth enjoy actively Trolling the galactic community.
Schlock Mercenary is probably the lone example to go for plausibility. Aliens with FTL travel probe the radio spheres created by inhabited planets. They're not even trying to decode the signal, just confirm the presence of one to study its most distinctive feature: Every radio sphere in this section of space is in fact a radio shell, hollow...
The Aliens (they are actually called that; they are from the planet Alien) from the Walkyverse are dopey pop-culture junkies who are obsessed with human television and movies.
Kila Ilo: Richard, Kila, and later, Dr Kao learn about humans via the internet.
The webcomic XKCD provides a helpful chart for identifying quotes and internet memes relevant that have reached specific star systems.
In Agent To The Stars, not only is this trope played straight, it's the basis for the story. Having received Earth's TV broadcasts, the aliens decide that the real power on Earth is Hollywood, and make their first contact with Hollywood's biggest agent.
In addition to their beliefs in the power of Hollywood, they're also unpleasant to human senses in many ways. In other words, they smell really bad and look really ugly. They would not get a good reception if they just landed on the White House lawn and their agent is intended to thwart the typical human reaction.
In Chaos Fighters II: Historical Chronicles-Beyond The Earth, this trope is used to explain why people in Vertrifo speaks English. They steal cable using magic, though.
The Soviet cartoon "An Old Record" is about aliens borrowing a few records to listen (and, through some strange technology, view, too).
Futurama, as usual, played with an old Sci Fi chestnut. Lrrr, of the planet Omicron Persei 8, 1000 light-years from Earth, commonly watches early 21st century Earth TV. His first appearance, with an invasion fleet, was because Fry thrashed WNYW's transmission console, cutting the signal when the Grand Finale of the Ally McBeal-esque program Single Female Lawyer was being broadcast.
Lrrr and his wife Ndnd are also apparently big fans of Friends... well, sort of, anyway:
Lrrr: This is ancient Earth's most foolish program. Why does Ross, the largest friend, not simply eat the other five?
The Junkions in the animated The Transformersmovie apparently learned English from watching Earth TV broadcasts. Which explains lines like "Steady as she goes, Bob! Snoopy visitors get mud in the eye, by and by! Film At Eleven!", and the battle cry "Destroy Unicron! Kill the Grand Poobah! Eliminate even the toughest stains!"
The key difference between Movie!Bumblebee and G1!Junkions is that the Junkions are capable of saying anything but choose to parrot TV and radio clichés with their own voices, whereas Bumblebee, who can no longer speak (or speak well, at least), hacks together bits and pieces of actual audio clips from the radio.
Kup is implied to watch Earth television as well, but less so than the Junkions. Fitting for his "old guy" image, the way he uses TV phrases (knowing only basic ones) is akin to an old man trying to use new slang.
Used very literally in Transformers Animated. Bumblebee and Sari pirate cable in order to watch illegal street races.
Scooby Doo And The Alien Invaders used this in an interesting meta-story way; an alien flower-child (due to having studied American pop-culture via its 1960's broadcasts and assuming it represented basic human behavior) became the Love Interest for Shaggy, who, having been Totally Radical when Scooby-Doo was originally on, and not having changed in the interim, was a perfect match.
Played with in Aqua Teen Hunger Force, in which the Plutonians are literally stealing the cable of Master Shake and company. They have a cable splitter patched through a Fargate to their spaceship, and use the "Universal Remonster" (a teddy bear with remote controls for arms and legs) to control it.
Not aliens, but demons: Neighbors From Hell shows that Balthazar was chosen specifically by Satan to go live on Earth as a human to get to the drill that threatens their home because he watches a lot of TV and therefore is already (theoretically) familiar with life on Earth.
Not exactly extraterrestrials, but in the depths of the Cold War some kids in Russia and Eastern Europe were able to receive communications from an alien culture - western pop music on Radio Luxembourg or the American Forces Network.
Similarly, due to technical and political considerations, the East German government decided it was impractical to jam West German TV and radio broadcasts, so most of East Germany (with the exception of the extreme northeast and southeast corners of the country, collectively derided as das Tal der Ahnunglosen or "valley of the clueless") had unimpeded access to what life looked life on the other side of the Iron Curtain.