Tight sweaters are often worn by ladies as a form of Fanservice.
A lot of the appeal is that they justoutline the figure (why "sweater puppies" is a nickname for a shapely chest) while barely actually showing anything. That makes wearing a tight sweater some of the most tasteful and acceptable fanservice and Fetish Fuel around (not to mention one of the few of the latter actually acknowledged in mainstream fiction). In the days of The Hays Code, once those sweaters caught on, they were almost up there with the heart-shaped neckline as a way of Getting Crap Past the Radar. Actresses who embraced this look were known as sweater girls, hence the trope name.
Not that a sweater can never be tacky. It's just if ladies want to look sexy, and the choice is between a halter top or an angora sweater, the latter is far less likely to have the wearer say, "Just because I'm wearing these clothes doesn't make me a prostitute!" It's more the "I'm way out of your league," kind of sexy. Particularly fuzzy sweaters can emphasize this by subtly echoing the glamour of a fur coat.
A special note as there is cultural differences, in Asia countries, school girls(and even boys) wear a sweater to cover up their developing figure, especially when they are growning to be more mature and starting to show differences in their bodies. So it is not uncommon to see school girls and boys wear sweaters when it is 30 degrees Celsius or higher. This has nothing to do with sexiness, and the wearers actually want the opposite. It is the authors that draw them to be much more revealing(as fanservice, obviously), but the hidden meaning is still cultural. Most of the Anime and Manga examples are actually girls that are more introvert, traditional and shy.
Also commonly worn by the Seemingly Wholesome '50s Girl.
Compare Fur Bikini, Happy Holidays Dress, Sexy Santa Dress, Homemade Sweater From Hell, Form-Fitting Wardrobe.
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Used quite often in ads, as a milder form of Sex Sells.
Anime and Manga
Shizuna-sensei of Mahou Sensei Negima! is often shown with one, which really accentuates her ahem maternal figure.
The girls of Archie Comics are prone to wearing the occasional tight sweater, especially Veronica.
At the end of TRON, Lora wore a white angora sweater, seen in the picture above.
In Lord Love A Duck, Barbara Ann (Tuesday Weld) wants sweaters so that she'll fit in with the 'in' crowd, and goes shopping for them with her father. The scene with her trying on the sweaters in front of him is rather disturbing.
Kristen Bell's character in Fanboys refers to her breasts as "Sweater yams".
Angie Dickinson wears the standard tight sweater with pointy bra in Sex Comedy (and very Black Comedy) Pretty Maids All In A Row. The sexually frustrated student who's the protagonist of the movie can't stop staring at her breasts, and she manages to twice accidentally hit him in the face with her breasts.
Janet Leigh was dressed this way in several films, including Holiday Affair and The Manchurian Candidate. A biography of her Holiday Affair co-star Robert Mitchum revealed that Holiday Affair producer Howard Hughes, who was notorious for his obsession with breasts and his predatory habits with young starlets, forced Leigh to wear an absurdly tight sweater after she rejected his advances.
Kim Novak, playing Judy Norton, is dressed this way when she meets James Stewart in Vertigo. This serves to contrast Judy's earthy sexuality with Madeline Elster (also played by Novak), the icy blonde that Stewart became obsessed with before her death.
In Are You There God? It's Me, Margaret., Laura Danker, the most physically mature girl in Margaret's class, wears a sweater and gets harassed. Freddie Barnett a.k.a. the Lobster asks Margaret why she doesn't look like that in a sweater, which Margaret does not think is funny.
Neighbor girl Lisa Hoober in Then Again, Maybe I Won't, Judy Blume's subsequent novel.
In William Goldman's Magic, the narrator flashbacks about seeing his love interest in a sweater and being aroused for the first time in his life.
"It must be remembered that the well-bred girl is always fully clothed in the office. The broken shoulder strap, the deeply split skirt, and the bare midriff are de trop in most businesses. The bright girl soon learns that these devices are not only in bad taste, but are not necessary.
"It is not skin area but contour that counts.
"A few simple experiments with sweaters, jerseys, and a slightly smaller dress size will bring pleasing and surprising results. One young lady who made a careful study of contour planning found that results were little short of breath-taking."
In The Drew Carey Show, Oswald's mom (played by Adrienne Barbeau◊) remembers how when she used to hug other boys, they tended to breathe faster. Drew admits it was her tight angora sweater, with the plunging neckline.
Millennium. A female friend of Katherine Black mentions that she was the "favourite student" of (male lecturer at college). Katherine replies: "I wore tight sweaters."
Christina Hendricks' character Joan Holloway on Mad Men deploys sweaters coupled with a variety of form-fitting but tasteful business skirts to devastating effect.
Once she tells a cleavage-flaunting new secretary to button up, saying that less is more if you know what you're doing. Said secretary ends up getting company partner Roger Sterling, Joan's former lover, to leave his wife for her, which says something about Roger's tastes.
Rachel Berry (played by Lea Michele) tends to wear these a lot on Glee. The show even lampshades it.
Jacob: Get those sweater puppies out of their cashmere cage!
Jack was often distracted by sweater girls on Three's Company, resulting in Freudian Slips on several occasions: "You need to separate the yolk from the sweater." "I just came to button up my coffee."
Spooks: In "Spiders", Zoe needs to get close to a particular target.
Tom: The kids' lives are in danger. (beat) So, wear a tight sweater tomorrow.
Tom: I've been watching him all day. He obviously fancies you. We need to build up on that connection.
Zoe: (sarcastically) Any particular sweater in mind?
Tom AND Danny, simultaneously: The blue one.
In an episode of Happy Days Richie, Potsie, and Ralph break into another student's college dorm to get back an incriminating photo he took of Richie; they discover a nudist camp magazine and get Distracted by the Sexy.
Annie in Community has a wardrobe that appears to mostly consist of cardigans.
Donna and Adrienne in NYPD Blue often wear tight sweaters. Diane usually wears skin-tight T-shirts for the same effect.
One episode of Growing Pains had teenage horndog Ben excited for his class field trip to the planetarium, which he intended to spend "picking fuzzballs off of Sasha Sorecki's sweater."
Inverted on Will and Grace: Will's been deceived into thinking he's going to be a Barney's model, and explains to Jack's boss that his broad shoulders and slim waist (IIRC) are perfect for sweater modeling.
One of Frasier's girlfriends was said to really fill out a mohair sweater in high school (specifically, he said the sweater in question was tighter on her than the goat.)
Arcuied returns in Fate/EXTRA as a Berserker, but still in this mode of dress (though a little stained and fraying at the edges). The overall effect is that of a suburban mother gone feral and eager to rip you apart with her bare hands.