I've always felt more comfortable around animals than people, so of course I would be drawn to animal and animal-like characters in media. (And there are a lot of them when you're a kid!) It didn't become anything of note until my teen years, however.
One day I was flipping channels on the TV when I came across a documentary about foxes. It was meant for children, but the information was new to me and I found myself watching to the end. A day or two later I was watching something on Cartoon Network when the main character suddenly held up a fox as a prop for a joke. The next day I noticed a random picture of a fox somewhere, and then it was like foxes were suddenly everywhere.
And yet, despite foxes being such a common animal with great importance in folklore and popular culture, I realized that I knew almost nothing about them. This felt like a problem that I needed to solve. I read everything I could about the creatures, both in my father's old encyclopedias and later on the internet when I got access to it, and the more I learned the more I started to identify with them. As a socially-awkward teenage nerd I really resonated with the idea of these small, solitary creatures struggling to get by on intelligence alone, without all the easy advantages given to their larger canine relatives. Now I began to imagine myself as a fox, and would often spend the last moments before sleep imagining the adventures of this other me. This idea of an "inner me that is a fox" would become a useful tool for exploring my identity.
While I consider this the start of my furriness, it would be many years before I actually joined the furry fandom. There was a lot of misinformation on the early internet that kept me away, and I won't repeat it here but I'm sure many of you know what I'm talking about. Then one day a friend of mine shamelessly held a brony birthday party and I decided that if he could embrace his weird interests so openly then I could at least admit mine to myself. I started lurking on r/furry, realized they were actually cool people, and was shocked to learn that the weird little fox people in my head are something other people have and that they're called "fursonas".