In the top left corner of the post-race menu, there is a prompt to press + to add (or remove) one star out of six. What do these stars do?
The stars allow you to save Highlight Reels as your favourites. You may then view these highlights in Mario Kart TV. Your favourites are available in the Favorites tab, by pressing R or ZR twice after opening MKTV. You may remove favourites from that tab by pressing -.