Currently, you can earn stars by collecting things. You can collect stickers, clothes, face features, furniture and colours. Each one of these things has a number of stars linked to it.   You are also able to win level XP by placing in the top ranks in leaderboards. 
The rarer the item, the more stars you will get. Whenever you obtain the first copy of these items, you will get the full amount added to your current star total. Any subsequent duplicates will not count.