What do the stars mean beside the trophy?

This is your ranking. When you complete a cup, you get a ranking based on your points and your time (and possibly other factors such as how many times you fell off or hit opponents with items). The best ranking is three stars, followed by two stars, one star, A, B, C, D, and finally E (I believe). The best ranking you've ever achieved in a cup is recorded for each cc, so if you redo the 50cc Mushroom Cup and get three stars, that will replace the one star.

Getting higher rankings isn't required to do anything in the game, but once you gain rankings of one star or better for every cup on every cc, the game will display one star alongside your name in races. Same for two and three stars.

the stars (or letters) that appears on the side of each cup is your ranking on that cup and it goes like this (from the best to worst): 3 stars, 2 stars, 1 star, A, B, C, D. I don't know if there's an E. simple explaining: the better you gon that cup, better ranking (Mr. obvious strikes again.), and when you finish all cups on every cc in the game and get at least one star on each cup, the worst rank you have will appear on the side of your name on multiplayer (if you have in almost all cups 2 stars , but you have a cup with only one, a star will apear on the side of your name, if you had 2 stars on every cup, you'd have 2 one the side of your name instead, the same goes for 3 stars.).

it doesn't affect the gameplay, it's just for status.