I woke up one day, dusted off my teletype and telneted into FurryMUCK, and registered a character. The day went not much differently than any other day, and has gone not much differently since then.
One of the most-cited "necessities" to becoming a furry is the picking of a "fursona". For some people, picking a fursona is an easy thing: "I like foxes. My fursona is a fox." For some people, picking a fursona is hard: "I like bats, but I like giraffes too." or "I'm strong and pack-oriented like a wolf but I also like foxes." (thus was the folf born). A large part of Furry involves art, so if you don't have a fursona it'll be hard to involve yourself in it, since you don't have an avatar. Imagine if Sully had to go around without his big blue demonic possession smurf.
On the other hand, a great thing about Furry is that you can be a furry without needing to make yourself an avatar. Some people like the art and stories. My mom likes the Blacksad comics because they're decent stories in a decent presentation. She's not a furry, though, because she isn't a furry. Some people wish they were animals (or their animal avatar). These people sometimes overlap into Therianthropy, which is completely unrelated. Some people like acting, and they like cute critters, so they buy (or make) costumes to perform in. The list goes on.
There are also conventions you can go to, to meet others and reportedly consume large amounts of alcohol with others.
Of course, if you meant what is it like to actually be a furry, a good place to start is to ensure that you grow your beard out very long. In fact, just disown every razor you posses. A good background in genetics and less-than-ideal morals also wouldn't hurt.
even though he tried to convince me to be a Kitty hehe. Anyone can be a furry, the hardest part is making up a fursona and name, atleast for me it was. Personally I think I kind of caved in to the furpressure and super cute cubby pictures x3