Star Points are a type of currency that you can see at the top right of the screen:Star Points score
With this new type of coin, you can get exclusive items from the in-game store (mainly character customizations), and you can earn them by participating in the Trophy League. In other words, as you increase a character's trophy count, you get Star Points, but not always. Sometimes you do, but in other cases, you may only earn ordinary Brawl tokens. However, to find out if you got Star Points, how many, and how many trophies are needed, select a brawler and tap on its particular trophy count:
Whether or not you can get these kinds of points is a bit random, as it is determined by how many trophies a brawler has won, and the reward for reaching certain achievements is also completely variable. However, the way to find out is already there.
In the old days, this was not the case, but there were Star Tokens. They were special coins that were obtained by winning games on Brawl Stars. When 10 coins were achieved, they could be redeemed for a Big Box for free, but with the last major update of the game, they have disappeared for a more Clash Royale like approach when it comes to the reward system.
Another important aspect: did you play Brawl Stars before the last big update? Did you win tickets for the weekend events? If the answer is yes, then surely Supercell has already notified you through the game, but those tickets are now Star Points.