Heh - well I don't know if you "become" a fur so much as realize you are. And that happens in several levels too. For most people, they identify with a certain facet of the furry/scalie fandom in that they like an anthro creature, or furry things, or relate better to animals than people - I mean, there's a 101 things that people define as "furry". Then after they recognize that aspect of themselves, they eventualy stumble upon the fandom and realize there's a term for it, and often that they're not unique (much to their relief).
For me, I realized I liked certain aspects of furryishness from the time I was 4, which was 1979. It happened when I was watching a Godzilla movie - Godzilla vs. Gigan. I was enthralled by Godzilla - a big anthro lizard and I just loved the look of him (and of course at that age, I didn't think the old 70's suits cheesy or anything). More so, there were a few scenes which showed his big footpaws, which I really loved.
Over the years that followed, other Godzilla films, cartoons and movies that had anthro creatures, kaiju and so on were something I just loved. Again particular weakness for footpaws which also became something I liked in humans too. It wasn't until puberty, for me at 11, that I really understood just how much the stuff was a turn on to me.
So, that continued for years into my teens and early 20s. By my early 20s, I was on the Internet (and the web was still kinda in its infancy then) and often did searches for such things, but never really found anything much. So it remained a private sort of thing that I figured I must be the only person in the world crazy enough to have a thing for anthro creatures, dragons, footpaws and so on.
Unofficially, I discovered the fandom in 1996, but didn't realize it. I had been looking for some pictures of T-bone and Razor from Swat Kats for a background I as making for someone and happened across some yiffy art work of them. I remember thinking "Whhaaa... people draw porn of this stuff!?" Rule 34.

But I never thought to look further because I was in a rush.
It was in 2000 that I found, quite by accident, the site BigLizardFeet which had not only a lot of footpaw stuff, but also links to all over the rest of the early fandom. So... shock and sort of happy awe that I was not alone in how I felt and there was a huge (and growing) bunch of people out there that liked this stuff.
THAT is when I realized I was "furry" by our more fandom definition of the word. Been gratefully so ever since.
