Those stars tell you how many KO's has that particular player performed on a single match round. They can be reset by getting out of the BRAWL mode, or by getting out of the game (that includes power shortages). The star count is not affected by rules or characters used, but it is affected by player selection. For example, if you are playing as "SMASH"and you switch to the name "IM#1", the stars earned by "SMASH" will be erased, and "IM#1" will start earning stars for the KO's. Even if you started using Peach when using the name "IM#1" and then you swithced to Ganondorf, the star count will remain on the last score registered when playing as "IM#1".