I grew up before the Internet was a "thing" but was always drawn to furry/anthro characters in books, movies, and they were the central figures in my own imagination/playtime. None of my friends were interested in that - when playing make-believe, they all wanted to be princesses while I wanted to be a dragon or unicorn. My parents gradually tried to wean me off of it, telling me that talking animals were "for little kids" and they tried to steer me towards things that most other girls my age were interested in - dressing up and learning how to do make up, and getting jobs as babysitters.
Not for me. I was basically made to feel embarrassed for liking the stuff I liked, but the stuff I was "supposed" to like was just boring to me.
I just kept it to myself, wrote stories or got lost in my own daydreams.
When I was just finishing high school I went to DragonCon in Atlanta, GA and met a friend from online there. She mentioned something about Omaha the Cat Dancer, something I'd never heard of before, but I looked into it later. I was like "wow, there are older people who do this stuff?" Looked around online more, discovered Max Black Rabbit, Zig Zag, the Sabrina web comic, and it just progressed from there.
I probably didn't actually consider myself a furry until about ten years ago, when I joined the Second Life grid and discovered that you could truly be ANYTHING there, including an anthro animal. I put together a bunch of different characters for myself and got into roleplay in a sim called The Wilds Sanctuary, a medieval world inhabited mostly by furries. It was really the first time I actually interacted with other people who were adults, and didn't make me feel like I was stupid for pretending to be an animal character.
That's pretty much what I've been doing since.