It is a way to distribute coins to more people. A donation, a giveaway of small amounts of the coin. It is called "faucet" because coins "drip" from like the way water can drip in small drops from a faucet. Most of the coins have such faucets.
Yes, it is a way to promote the coin. It is a way for a guy who has never heard of the coin to start collecting it. It is way faster to mine the coin, but for people that have no hardware to mine it, faucets are good start.