Okay, here goes...
The first, more cut and dry, reason I like anime are the great characters in anime series. They have much more depth than characters in most American series (live or animated), and although they are animated I can often identify with them on at least some level. And as others have already stated, the plots and storylines in anime are almost always deeper and better developed than those in any American programs.
And I agree, WhiteBlaze, the way anime series build on a plot throughout a 26-episode (or in some cases longer) series gives so much more room for development and really is appealing to me. I love to be left hanging and eagerly anticipating the next episode, trying to figure out what might happen next. I personally prefer anime series to one-shot anime movies for this reason.
The final reason I like anime is less cut and dry, and perhaps a little hard to explain. I could just say I love the art style and visual appeal and leave it at that, but it's something more than that. The visual characteristics of anime resonate with me somehow on a physical level. Now before anyone goes getting weird ideas, let me explain. I think my physical brain is very attuned to spatial input. I've always enjoyed the crisper, cleaner, sharper (more geometric?) lines of anime to the softer, less angular lines of older American animation. But I've always preferred animation of just about any kind to live programs of just about any kind for as long as I can remember. I never used to like programs like "The Great Space Coaster", live kids' shows with puppets. I wanted to watch cartoons. And when I first saw the style of Voltron I just about went crazy. It's the same way when I draw, I do better with things like buildings or mecha than people because they have more angular, geometric lines. I also liked geometry a LOT better than algebra in school--go figure. And with anime specifically, I like the art style of newer shows like CLAMP series or Slayers better than 80's anime with its generally softer lines, for example Macross/Robotech (the original series). So that's my long-winded way of saying that I think I have an actual biological propensity toward anime because of the way my brain is wired.