After a race, at least a Grand Prix race, you can hit x to star the race. What's unclear is what that does.
The star after completing the track is for the race, not the map. When you star the race, it's highlights are saved in the Mario Kart TV (MKTV) section. Your 12 most recent races are stored, along with 6 additional slots that you chose by using these end-of-race stars.
You can view, edit, or delete the saved races by going to the MKTV section at the bottom of the main menu, going to the race you want to delete, and hitting Y.