The first Japanese opening theme was fairly expository itself.
The original Mobile Suit Gundam did this to great effect declaring the Gundam has the only thing standing between the Earth and utter annihilation at the hands of giant enemy robots
The English dub of Sonic X has one to an extent: "Got ourselves a situation / Stuck in a new location / Without any explanation..."
GaoGaiGar's opening is not so much an explanation of the show's premise as an anthem directed towards the heroes. ("Rage, steel cyborg! With a red mane and a golden arm! Shining G-Stone, to protect the hopes of the world, stand up now!")
As a secondary invocation of the trope, the collective image song of the GGG Mobile Unit (HyoRyu, EnRyu, FuuRyu, RaiRyu, Volfogg and Goldymarg), Saikyou Yuusha Robo Gundan, describes the characters quite neatly, one per verse.
Years later they did a version for the French Dragons KouRyu and AnRyu and their combined form TenRyuJin, for FINAL, the aptly titled Saikyou Yuusha Robo Gundan, Ladies.
The theme song used in every adaptation of Cutey Honey is a variation, describing not the plot but the protagonist, and fitting in quite well with the fanservicey nature of the series as a whole.
Most classic super robot shows (or any new show trying to emulate them) will feature songs that describe anything from the robot's weapons to their battles, often overlapping with Bragging Theme Tune. For example:
"I wanna be the very best! Like no one ever was!/To catch them all is my real test/To train them is my cause/I will travel across the land/Searching far and wide/Each Pokémon to understand/The power that's inside!/Pokémon, Gotta catch em all!/It's you and me/You know it's my destiny!/Pokémon!/Ooooh, you're my best friend/In a world we must defend!/Pokémon, gotta catch em all/A heart so true/Our courage will pull us through/You teach me and I'll teach you/Pooo-keeee-moooonnnnn!/Gotta catch em all, gotta catch em all, Pokémon!"
And who could forget the opening to the original Pokémon Chronicles (Raikou, Legend of Thunder in Japan)? The opening sounds like a freaking advertisement when the announcer reads, "Old friends! New adventures! Never-before-seen stories from the world of Pokémon!"
Just the US dub? Japan does it too y'know. The title of the Japanese season 1 opening outright translates as "Aim to be a Pokémon Master".
The English dub of Amazing Three. "Spacemen with a mission, you must make a very big decision..."
The English dub of Sailor Moon has a song about the title character "Fighting Evil by Moonlight" and "Winning love by daylight". It matches up... for the first season. Between having found her true love, her powers no longer being "new", and any Senshi that shows up after the first arc being left out... The original didn't have this and the song was more about love — specifically finding your true love (or "Miracle Romance").
Anime Theme Songs are rarely directly related to their shows. English dubs for children's TV tend to invoke this trope more often.
The (to be dubbed) 5th season, replaces the standard opening theme, "Moonlight Densetsu", with "Sailor Star Song", which was written just for the show (the lyrics were actually written by the Mangaka of the original manga) but still isn't really this.
The dub of Ultimate Muscle had a quite catchy, but blatantly expository one. It manages to be less of a Spoiler Opening than the Japanese one, due to the increased speed of the animation making it more difficult to pick out all the characters shown.
"Ahhhhhhhhhhhhhhh / Searchin' for the Ci....ties of Gooooooold / Ahhhhhhhhhhhhhhhh / Someday we will find... the Cities of Gold."
The English dub of Monster Rancher. "I was transported / to a faraway land / in a world where monsters rule. I played the game like an ace / now we're in this place / to save the monsters from the evil Moo."
How can anyone over 30 forget "Off to outer space we're leaving Mother Earth, to save the human race, OUR STARBLAZERS!"
The theme to the English dubbed version of 8th Man combined this trope and Never Trust a Trailer, as the theme and (American animated) opening sequence described 8th Man fighting aliens (which he never does) and flying (which he can't do).
Ulysses 31: "Ulysses! Ulysses! Soaring through all the galaxies! In search of earth, flying into the night!"
Digimon Adventure, while being one of the better dubs in existence, suffers from a particularly repetitive, if kind of catchy, theme song.
Digimon Frontier on the other hand, has a more vagueish, yet still quite expository theme.
''Look to the past/As we head to the future/To reclaim the Digital World/With faith in ourselves/And trust in each other/We live by the lessons we've learned/As we work toward one solution/Though our Spirit Evolution/Digimon!
The opening themes of the 2012 JoJo's Bizarre Adventure series have had these for the first two parts so far. The theme for part 1 makes subtle allusions to the plot, but the part 2 theme is pretty much an ode to the protagonist and his struggles.
Deadman Wonderland has "One Reason", a song specifically made for the show based on the existing song Black Hearts and "Dollar Signs" by Fade. "I will be your Deadman, with nothing but this blood on my hands, trapped in your Wonderland..."
As is common with Mon series, the English intro to Yo-kai Watch is exposition. Part of it is a rap and the other part is singing. The rap even begins with "So sit back. Here's their story".
Laura's gone to school, let's go to our Ham-Ham clubhouse / We can fix their troubles, just be quiet as a mouse / Watch out for those cats, you know they're smarter than you think / But if we work together, we can make their plans sink!