My grandfather starts his day by going to a public bath.
Give a coin and get hot water. So you hand over a coin and then you get hot water in the form of a bath. It's a bath for anyone willing to pay, making it a public bath.
This is a jukugo word that uses the on'yomi readings of the kanji. You don't know the on'yomi reading of 湯, though, so just think about how when you get in the public bath, you're in Tokyo (とう), because there are a lot of public baths in Japan, and Tokyo is the most well known city there.