You can win stars in several game modes. These stars fill your star bar in the home screen and augment your general progression level in QuizDuel. Your current progression level is displayed next to you avatar as a number in a circle.
For each filled bar a present awaits you. Tap on it to receive your reward!
When you reach certain levels you also rise in rank, which is visualized by a frame around your avatar. If you do not want to see this frame displayed simply unequip it in your avatar editor. You find this editor by tapping on your avatar and then the paintbrush next to it.
Additionally, stars open up the chat functionality for you: once you've reached level five, you can chat with other players.