ClockWork said:
I just don't see it. If anyone can sway my thoughts to the contrary, I would more then welcome it.
First things first, I should mention that I got into anime in the mid 90s, heyday of shows such as Sailor Moon and Ranma 1/2.
At that time, anime was this strange, alien thing that was unlike any of the boring crap on American TV. My teenaged self was enraptured - particularly by shows featuring ass-kicking women doing cool stuff.
For me, anime peaked in 1997-2003, with shows like Revolutionary Girl Utena, Noir, Escaflowne, and a few others.
The last five years or so, I've only found a one or two that I like, and none as much as those shows from the previous decade. Also, many Western shows in the 2000s adopted the best, most interesting parts of anime (such as Joss Whedon, the creators of Power-Puff Girls, and others) thus eliminating the need for importing them directly.
I think it was just something different. As Movie Bob points out, the 90s kind of sucked. Very boring and drab compared to the decades surrounding it. Meanwhile, Japan was hopping with all sorts of cool artistic ideas. Now, ten years later, America, England, Canada, Australia (not to mention Europe) are producing interesting stuff, and Japan is, well, kind of dull. Much of the more recent animes are rehashes of older, better shows.
Also, back in the 90s, only the very best animes got exported to the west (based, of course, on the particular biases of those doing the exportation). Now, pretty much everything comes over, good or not.
So, while I'm still an anime fan, I don't watch much of the new stuff coming out. I stick to my older series and enjoy them, and I watch western shows inspired by anime, and... yeah.
tldr: I think Anime was better when it was new and different. Now that it's been incorporated into western culture, much of what made anime special has been subverted by western authors. Which is a good thing, because it attempts to preserve the best parts of anime while getting rid of the bad bits. Not that everyone agrees on which is which all the time....
Edit: if you actually want to get into anime, try something older. 80s or 90s older. You'll find some of the best stuff then (including fan favorites like Cowboy Bebop).