I think it probably stems from a sort of personality-induced thing, like SubLet said, combined with (especially re: DuckTales, Mickey Mouse, Looney Tunes, etc) a common childhood facsination with "what if I were an animal"?
Probably the best example of personality furries are the Daemons in Pullman's "His Dark Materials" trilogy. They're not even vaguely human-shaped, but they're the souls of their respective humans and can talk.
Most kids like animals, and wish that they could talk to them, or could be like them. Furries fulfill this fantasy.
Historically there's a lot of facsination with anthropomorphic animals. Examples: native americans and spirit guides, Egyptian gods with bodies of humans and animal heads, Greek myths involving animal transformation (Gods, Odysseus' crew, etc.) Modern day furries are usually linked to the kinky thing, but really, that kind of thing has been around for a long time (see: the more ew-provoking Roman arena events)
For most people, especially in webcomic-land, cartoon furries are probably a combination of "I like to draw in this style," "anthropomorphic animals are interesting," and "they're so much cuter than humans."
But really, Lastcall, I think it's kind of like jazz. If you have to ask, you'll never know.