I'm going with the part where it's only 3 syllables as to why it's won out over 'japanimation' and 'japanese cartoons'. Anyway, as has been said, we're co-opting a Japanese word that comes from an English word to describe something which tends to be more specifically Japanese than, say, French cartoons are discretely French. If the animation style, voice acting, plot elements, action points, themes, and virtually everything else tend to have things setting them apart from western things, just calling them cartoons would lose something of description. It's an entire subculture, is one extra word too much to ask?
Linked for hilarity: http://uncyclopedia.wikia.com/wiki/Anime