There are quite a lot of bitcoin faucets on the Internet. They give away bitcoin for free, but where do they come from? Won't the faucet operator lose money?
Advertising. The amount you receive from faucets is so minimal that the amount they make from your visit from you seeing ads is more than the amount they'll give you.
It's rare for a faucet to do it simply to solve CAPTCHAs, as most CAPTCHAs are generated randomly, and trying to find an answer to all possible graphics is virtually impossible.