This happens sometimes with couchsurfing as well. People are generally good and trusting.

That struck me as odd too -- is this a normal use case for airbnb?

Yeah. The one time I used it, my brother sent a phtocopy of his driver's license to the apartment owner and she sent him the keys in the mail. I was a little nervous about it all working out, but all the hotels in the area were booked or were $600+/night. It ended up being a pretty good experience for us, but it did require both sides to be pretty trusting.