A few reasons.
First, animation can do things virtually no other visual medium can. The diversity of possible topics, the creativity and depth in which they can be explored, the absolute freedom of any visual representation of any idea, no matter how outlandish or high-concept it might seem… that’s awesome. Many anime could not possibly be translated to any other medium and maintain what makes them great.
Second, anime tends to focus much more sharply on the interior lives of characters than many other media, and I really appreciate that. I enjoy a good plot as much as the next person, but my favorite anime are all about subtle character moments and explorations of what makes people tick.
Third, the best anime often offer combinations of strengths that surpass almost any other media – I’m thinking here of moments where a long-building plot, a strong visual aesthetic, a slowrolled character turn, and a fantastic musical accompaniment all come together to make something incredibly unified and powerful. Moments like (Eva spoilers) this or (FLCL spoilers) this are why anime strikes me so powerfully.
Fourth, I really like “broken” works – works that are flawed in some obvious ways, but also compelling or deeply personal in other ways. There are many anime that could never, ever attract a mainstream audience because of some glaring flaw in immediate storytelling, or a necessary suspension of disbelief, or some strange visual choice – but that clearly demonstrate the singular vision and mind of a quirky and dedicated creator. I prefer my art un-streamlined, and while I can get plenty of this in music and film, anime combines this with the ability to tell a story over a larger time frame and the strengths I’ve already mentioned.
Fifth (I know, sorry), I like the pacing of many anime more than Western media – I like how so many scenes are built out of peaceful establishing shots, how so much is left unsaid in slow conversations, how the environment can often become as much of a character as any single protagonist. The town of Mabase is an intractable part of a show like FLCL – the environment does most of the telling in 5 Centimeters Per Second. To me, this really cements these stories, and makes me feel more a part of the character’s lives than more direct action and telling ever could.
Finally, I just like it. At this point, I’m programmed that way – an adolescence of Toonami, a young adulthood dominated by the visions of Gainax, Watanabe, and Miyazaki, and all the other shows I’ve seen and manga I’ve read along the way have basically conditioned me to feel incredibly comfortable and almost safe watching any new show. There will be things I’ve seen before. There could very easily be a high school that is not all that it seems, containing a somewhat detached student, sighing as he looks out the window from his window seat second from the back. And that’s okay. That seat’s a pretty fun place to be.