It's an experimental anime from Gainax and Production I.G. It's pretty whacked-out. The story it's telling is actually quite straightforward, compared with other Gainax shows that go completely off the deep end into the land of "you have got to be kidding me" in the closing stretch. It doesn't disappear into metaphor and random free-association, which will be because Anno Hideaki didn't work on it. The plot means what it says and everything that happens can be taken literally.
However it's also so crazed and self-aware that your brain's unlikely to get the message that these events could ever be taken at face value.
We begin with an obnoxious twelve-year-old, Naota, and his brother's abandoned girlfriend, Mamimi. They're under a bridge. Naota's doing his homework and Mamimi's practicing baseball, but then a woman flies down on a motorbike and hits Naota with an electric guitar, causing a horn to grow on his forehead that will soon-ish become about a thousand times bigger than Naota and turn into a giant robot. See what I mean?
Crucially, all this is fun and funny. The guitar-wielding biker, Haruko, is an entertaining nutcase and that initial collision scene had me laughing out loud. This anime isn't a comedy as such, but about once an episode it would make me laugh anyway with character or sight gags. The animation is willing to do Tom and Jerry style deformation with realistically drawn characters, or else South Park. Seriously. Episode five briefly does South Park. There are sometimes subtle in-jokes about Lupin III, Neon Genesis Evangelion and Gainax's Daicon mini-movies.
The director has said that he tried to "break the rules" of anime when making FLCL, e.g. using a contemporary Japanese band for the soundtrack, or copying the style of "a Japanese TV commercial or promotional video" to be "short, but dense-packed". Normally when someone says they "broke the rules", this means they're full of it. Here, I'd say they managed it.
It has emotional weight, but this tends to be slipped in obliquely. Ninamori, the mayor's daughter, is the only character whose issues are delivered straight. I liked her episode. Everyone else's stories are mixed up with gags, wacky SF and/or giant robots, making it easy to miss some of the emotional content in the melee. The arson, for instance, is implied so clearly that you'd have to be an idiot to miss it... and then the series quietly drops the subject and never mentions it openly again. The audience is being expected to keep everything in their own heads. This show is hard work to watch. There's also religious imagery when a robot with a halo and wings ascends into the heavens. Yes, the scene is that explicit.
There's a lot of metaphor and intriguing intellectual content. The most interesting, for me, was Naota's insistence that he lives in a boring place where nothing happens, which leads him to assert that Haruko's electric guitar causing giant robots to sprout from his forehead, for instance, is an everyday event and nothing special. More generally though, the series would appear to be about growing up and about the differences between children and adults. There's lots of sexual metaphor, e.g. Naota's phallic horn, representing the confusing and embarrassing physiological changes of adolescence, but the only actual sex takes place behind closed doors and involves adults being irresponsible life-wreckers. The plot might be driven by the universe's most irresponsible sex drive, although there are other explanations too and it depends whom you believe.
This was an original show, created for this medium. It wasn't adapted from a manga, although a manga has since been written that's apparently darker and more graphic. There's also a three-volume novelisation. This anime left me a bit nonplussed, but in a way I think I might have been better prepared for it if I'd never seen any Gainax anime before. I knew how fond they are of psychological metaphor and non-literal narrative, so I think I kept my distance a little too much. This show certainly has plenty of metaphor, but it also has a plot that means what it says. The middle episodes tell strong character-based stories that reward viewers who take them literally and pay attention to the subtext.
It's bonkers and potentially off-putting, especially when you reach the final episode. They don't lose it, unlike His and Her Circumstances or Diebuster, but you shouldn't expect to be spoon-fed either. However it manages to hold itself together, despite constantly threatening to disintegrate under its own centrifugal force. It's attention-grabbing; I'll give it that.