The game will never start automatically. The game will start when both player will be seated, and when they will both agree to start. This is to make sure that both players are really ready when the game starts.
There are two seats in each room. When a seat is available, click on its button to seat there. If you want to leave the game, you don't need to leave the room. Just click a second time on the button and the seat will become free again.
Once two players are seated, the game doesn't start. Both players must be ready to start the game. You need to click the button "Ready to start". You will see the word "ready" written on the seat's button. When both players will be ready, the game will start immediately.