Well, I always thought of the whole, anthromorphized thing to be really cool. I love the thought of humanoid animals and wished I was one myself from a young age. Then it was mostly things like a t-rex and the sort, beastly things I could pretend to be to scare my little brother. Then it sank back for a few years. I really discovered my furriness when I started hanging out with my group of high school friends. A large amount of them were furries, I remembered my thoughts years ago, and things resurfaced. Only now I was some manner of feline. Now I find my fursona to be that of the cat-boy, though, things can change.
Also, as everyone has said, you don't become a furry overnight, it's something that you have for a while that develops into, well, an alternate side of your personality pretty much. During the day you could be boring cubicle slave, but once you get home, you're a lovable fox that everyone wants to glomp!