Thought I'd pop up to answer the question posted in this thread as I like anime, well... actually anime is probably my main hobby.
Why do I like anime? My enjoyment of anime comes from a deep rooted love of animation even to such years as my middle teens, I discovered anime when I was around 12 years old and have been hooked on it ever since. It's even managed to secure the country of Japan a place of interest in my mind. My original love was wrought from Pokémon and Dragonball Z, some older anime fans hate the 'Pokémon era' fans as they came about to encroach on a hobby which was very specific and overall very centralized. Anime was difficult to get a hold of back then, or so I hear.
I love the action, I love the romance, I love the drama, I love the sci fi, I love the comedy, I love the hentai and so on and so forth.
As you can undoubtedly notice there are an awful lot of genres in there. The reason for this is because the Japanese think of anime as a medium and not a genre as the rest of the world seem to. Asking the question: "What's so good about anime?" to an actual anime fan and you will often get the response: "Well... what's so good about TV and film?"
Anime to the Japanese is actually all forms of animation. American cartoons would be anime to them, or so I understand it. However anime is, to the Western world, specifically animation which comes from Japan.
Not every anime will meet your tastes, especially the shite they usually show on TV. If you want to see real anime you're either going to have to get into buying it or fansubs. Fansubs are a pretty grey area in Japan, technically it's illegal but it's outside of government law. It's up to the owner of the media to prosecute and take action and for the most part they don't bother because fansubs are a seemingly great way of getting their shows picked up by American distributors. Why... Mezzo DSA only had to be aired for about 7-8 shows before ADV picked it up, though I credit that to Mezzo Forte, a hentai made by the same people set in one of those 'alternate realities' It's actually a fairly famous hentai.