I have no idea, since, apart from a very short list that always comes up in any discussion (the works of Miazaki and Grave of the Fireflies) the vast majority of it is truly awful animation.
To me it represents two genres you’ll never see on US TV. Fantasy comedy and Science Fiction comedy. Plus I like the bizarre humor. I also like watching shows like “He is my Master” and wondering what weird drugs the Japanese are taking.
Searching for “He Is my Master” is extremely NSFW!
To the extent that anime* is “popular” in the West (and I would say that it is popular to a degree worth remarking upon; it certainly has more of a following in the U.S. than most other types of foreign film), I think timing and lack of competing product both have a lot to do with it.
After all, while anime has been around and maintained a niche fandom for years, it really blew up in the mid and late 90’s, when VHS tapes became ubiquitous and cheap (and eventually replaced by even cheaper DVDs) and there was really no American equivalent of the types of thing people liked in anime. Most of the cartoons produced for adults in the US at that time were basically sitcoms. There was the odd sophisticated action/sci-fi cartoon like the 90’s Batman and Superman cartoons, but because they were marketed toward children those cartoons couldn’t feature anywhere near the type of material that anime can. The only thing we had that was comparable were the various MTV cartoons like Aeon Flux and The Maxx, and those didn’t last long.
Equally important, I think, is that huge portions of the country spent the 80’s and 90’s playing the crap out of Japanese video games. Lots of people grew up consuming tons of Japanese entertainment in the form of video games, so I don’t find it surprising that they seek it out in other forms.
And of course post-Pokemon and Yu-Gi-Oh!, Japanese cartoons are pretty much ubiquitous among a generation of kids.
*I don’t like this word, either. I think “Japanimation” is at least as bad. I like “cartoons from Japan.”
I just came in here to echo what Big T said. I remember growing up as a child, there were two cartoons that I would watch after school: G I Joe and Robotech. I quickly lost interest in G I Joe. Hell, I remember missiles being launched at choppers / planes and just as the missile hits, the characters would all jump out and parachute to safety. In Robotech one of the main characters was killed in an accident and I thought that was awesome! It was a way more serious show. Been watching anime ever since.
Aren’t the Japanese themselves the world leaders in embracing–even fetishizing–products of other cultures (random examples)?
I think of it in manga terms–& it is after all, Free Comic Book Day–but anime ties in. I think this is a big part of it:
There used to be a wide variety of comics in this country. But the cartooning business in the USA (& to a degree in the UK) is moribund. The romance comics are gone, the high fantasy comics are only done in small-press runs, there’s not even any new space opera with any visibility. The American “comic book companies” that could have produced this decided instead to focus on fanfic about hoary old trademarks (Archie), comic book shops with a small customer base (Dark Horse, Image) or both (DC, Marvel).
But it was relatively cheap to import stuff from Japan, with its fecund manga/anime industry. So a few companies did.
Also, Japan does the intersection of print cartoon & animated cartoon really well. Ever see a faithful adaptation of anything in American cinema? Ever see the original creators on art direction? But in Japan, it’s normal for the creator of the manga to also be creator of the anime. They market them together. In the USA, animation & comics are mostly separate–even when trademarks cross over, they’re reinvented. So Japan does marketing better.
It’s also important to note that it goes the other way. Japan gets a lot of our animation & live-action stuff imported as well.
A) Generalities aren’t absolutes.
B) Saying that you dislike animation is equivalent to saying you dislike live action. Unless you truly mean that you dislike the visual platform, more likely you’ve simply been shown a series or three that you didn’t like.
As an exquisite (for me) aside, I’ve finally found out where I picked up that toy robot that I had as a kid, that I used to carve bits out of my bed with. It hadtiny hardened plastic horns that made short work of the bed, my parents wondered where I was getting the knife from 
But to the OP, in the UK at least there was perhaps already an opening for that kind of show after all those Gerry Anderson shows with a similar technological bent about them. If the same kids/teens were interested in both, I wouldn’t be surprised.
I’d agree that the different culture is part of the appeal. I have two siblings who are into anime, and they also just really like Japan and the culture. I never got into it (except of course for Miyazaki) but I can understand it.
I went a different way and love Indian movies. Most of the time I would rather watch a Hindi film than an American one. The different culture is part of the appeal.
You weren’t responding to me, but I would also say I don’t like anime, and it’s not at all equivalent to saying I don’t like live action. I like my entertainment more grounded in reality than cartoons will ever be. I used to like several animated shows back in the day, but I grew out of it. The only animated show I can stomach nowadays is South Park. Everything else just seems so childish to me, even the stuff targeted to grown-ups. It’s too nerdling-in-dorm-room for me.
I was exposed, initially, to Robotech (which didn’t age well upon recent viewing. 
) it was shown early early in the morning on an upper-cable channel (actually, off satellite).
It was the first cartoon I’d seen where characters actually died.
I was re-exposed with Akira. You’ve gotta understand that the gulf between Scooby-Doo, Hanna Barbara, and Walk Disney and Akira was VAST.
It’s different now, but the seed was planted. It’s nice to be able to cherry-pick stuff like Cowboy BeBop.
What, never?!
I’m saying I don’t like Japanimation, specifically. I’m sure there are whole worlds of it that I’ve never seen, but what I have seen has a stylistic consistency that I find unappealing.
Plus there are only some many times I can see “running” that consists of two rapidly alternating frames, “talking” that alternates between mouth inhumanly wide open and mouth shrunk to a dot, characters that are identical but for hair colour, etc. I guess the quality stuff doesn’t make it over here as much.
Some American cartoons that are influenced by Japan are interesting. I liked Batman Beyond, for example, and I did hear of an anime that was an adaptation of The Final Countdown, which sounded interesting, since giant robots and mysticism do nothing for me.
No, never. 
Anyone who can can look at something like Pokemon and find simple relative comparisons to something like Serial Experiments: Lain must be operating at a higher level than myself. The former is an unabashed kid’s adventure show and the latter is one of the most complex meditations on human consciousness and artificial intelligence that I’ve ever seen out outside of an academic text.
That said… I’m about to look like a hypocrite…
Not a few posters here have noted that they were introduced to anime at a young age, and I’ll include myself in that total. Like others I watched Gi Joe and Robotech at the same time, in the late 80’s and early 90’s, and while the latter hasn’t held up necessarily, for a young teen the differences are more than noticeable. The level of moral complexity introduced in something like Robotech were groundbreaking, and from there I started watching things like Tenchi Muyo, Evangelion and Crying Freeman as I grew older. I never stopped regarding animation as a viable and serious dramatic medium.
But I have to say, in what’s generally regarded as anime there is a certain level of unifying stylistic and cultural choices that do present a coherent whole. Giant mecha, magical school girls, and moe dramas do not interest me… generally speaking. I can understand where some are coming from in rejecting the medium as a whole. I do still watch particular series here and there, but I don’t consider myself a fan generally.
And that’s not even getting into technical criticisms of the animation itself. A lot of it does cut corners in the same ways a lot western animation does, and for the exact same reasons. For someone interested in animation as an artform in and of itself I can see where they might not like it. But for a lot of us it was a bridge that didn’t exist otherwise.
I’d add to this that a lot of anime fans/Japanophiles seem to believe in a sort of fantasy Japan that is very different from the real country. I used to live in Japan (briefly as a child, and for a longer period as an adult), and when I meet anime fans they often want to talk to me about it. Some have actually made a serious study of Japanese language and culture, but I can think of a number of times when anime fans have said things about Japan that are not merely incorrect but essentially the opposite of reality. For instance, I knew a girl in college who thought it was common for Japanese high school students to live in their own apartments without their parents*. A poster here once tried to tell me that Japan was much more open than the US with regard to homosexuality. I’ve also encountered people who seemed to think that the entire country of Japan is just like the Harajuku and Akhibara neighborhoods in Tokyo.**
I don’t think all anime fans have an unrealistic image of Japan, but for a fair number of them I think this notion of Japan as a sort of geek paradise is a big part of the appeal.
*I pointed out that Charlie Brown’s parents never seemed to be around either, but this doesn’t mean little children often live alone in the US.
**In fairness, I’m sure there are Japanese people who think all of America is just like Manhattan.
I’m the same way. In fact, I watch very few Japanese cartoons outside of the Ghibli films. My problem tends to be that I find the English voice acting in most Japanese cartoons to be bad to the point that I can’t concentrate on what I’m watching. I much prefer Japanese-language tracks with subtitles; the Japanese voice acting could very well be just as bad, but since I don’t know the language I’m blissfully ignorant and can just ignore it and read.
Anyway, I still sort of bristle at the “It’s all giant robots and school-girls” criticism, even though I do understand it, in the same way that I’d rake issue with somebody generalizing American animation as being all comedy-driven and centered around talking animals, even though that’s also true to an appreciable extent.
What I generally find so bad about the voice acting is how long it takes them to say stuff. This isn’t made any better by the subtitles. And the voices (in my mind) don’t match the characters as well.
A lot of girls love manga and anime, and are really adept at drawing the characters. Unlike superheroes, who leave most women pretty cold, manga and anime seem to really speak to a lot of (somewhat geeky) girls.
Japanimation? Are cartoons from Germany called Germination? 
I used to enjoy anime quite a bit when I was younger. Like many people here I cut my teeth on shows like Robotech, Starblazers, and a few others I can’t remember the title of right now. Of course I was also a fan of shows like GI Joe, Transformers, Thundercats, Bravestarr, Smurfs, and all manner of cartoons. As I grew out of my teenage years I started to look at anime with a more critical eye. For the most part I just don’t think it’s any better than anything we have on American television. I like Cowboy BeBop a whole lot but I don’t think it’s any better than Magnum P.I. I used to think that there might be cultural differences that made it difficult for me to appreciate it, but, honestly, I think it’s just that most of them aren’t very good.
When I worked at a comic book store in Dallas the vast majority of our adolescent teenage girl customers were coming in to buy the manga books. If you go to an anime convention in the states (A-Kon in Dallas) you’ll see a pretty high percentage of women and girls attending. At Barnes and Nobles I typically see girls huddled around the manga section rather than boys. My university has an anime club that’s mostly young men but there seems to be a sizable minority of young women. (I know anime and manga aren’t the same.)
I don’t really know why it’s popular. I would guess that anime is popular, in part, because it’s different from what we typically see here in the United States. It’s a niche popularity to be sure but it’s a niche that has existed since the 1980s at least. I suspect most people outgrow it though (that’s no insult, I still watch cartoons).