When I first played this a few years ago, I didn't know too much about Batman. I didn't have to. This game serves as an excellent introduction to both the licensed LEGO titles by Traveler's Tales and the various interpretations of the Batman mythos from DC Comics, and if you're a fan of either, I'd be rather stunned if you didn't enjoy this.
It's your standard Batman plot. This one is not ripped from any existing film or comic, but is instead a new situation made to send you through as much of LEGO Gotham as it can, as both the Dynamic Duo of Batman and Robin shutting down the evil plans of Arkham Asylum's escaped supervillain denizens, and those iconic villains themselves going out to cause trouble. It's a very bright and colorful Gotham, though still quite dark. I guess it'd be more accurate to say it's a high contrast Gotham, and this leads to a rather gorgeous visual design heavily influenced by the Tim Burton films, a place that's beautiful when it's not being creepy as all heck.
It's much more the campy, silly kind of Batman than the darker Dark Knight of late (Arkham Asylum totally nailed that shortly after), though like I said, a few places are still certainly creepy. There are some changes for gameplay's sake, such as all of the villains being shipped off to Arkham Asylum (instead of some going to Blackgate Penitentiary, as Arkham is a hub level). Instead of constantly changing characters, the Hero missions have our heroes constantly changing Suits found within the level. Some are really cool, like the Glide Suit allowing you to soar or the Attract Suit allowing you to vacuum things (it's oddly entertaining), while others are just kind of a hassle or quickly forgotten about, like the Heat Protection Suit (good luck remembering it's a thing) or the Magnet Suit (have fun walking up walls very, very slowly every other level). You're able to switch between these as though they were separate characters in Free Play, and the villains don't have such gimmicks, so it kinda evens out in the end. You're able to unlock suit upgrades, and many of them, like speeding up the intolerably slow Magnet Suit to tolerable speeds, feel like they should've been available from the start.
The levels are very distinct, and for the most part, I would say very fun. This was, however, just before they started handling vehicles properly again in the LEGO games, so there are some completely infuriating moments where you have to tether your vehicle to mines and slam those mines into things, but they don't and they just slide around wildly until enemies blow them up and you have to start all over. When this is not the case, they are entertaining and explosive action sequences, but dang are they a pain to 100%. (Admittedly, I still boot up the game exclusively to play the Batmobile level, because you get to drive the Batmobile and just tear through the streets of Gotham.)
So, fans of the LEGO games, this was the game that set the par for the course, bringing in targeting mechanics, 32 levels, and a long list of increasingly obscure characters (Batman fans probably won't believe "Joker (Tropical)" made it into a "kid's game"). Fans of the Batman series, you'll find that much as the game takes liberties for the sake of humor or gameplay, they did really do their research first before intentionally making these choices. And you can play as Bane, Alfred, Batgirl, Nightwing, Hush, and one extra at 100% you're bound to geek out a bit over. Bios for each of the villains appear in the levels where the heroes fight them, so if you're new to the series and know very little about it, you'll come out quite well-learned.
All in all, it's fun. Campy, creepy, and stuffed to the gills with little nostalgic nods, atmospheric combat, and iconic characters. For once, it's exactly what it says on the tin. LEGO. Batman. There you have it. If that sounds good, go for it.