Ultimate leagues are a competition for ultimate cars. You need to compete in groups for promotion during a time-limited round. You earn score points for each victory in fights. At the end of the round, top performers are promoted to the next league and low performers are demoted to the previous one. There are currently 7 leagues in the game. 
Golden tickets are your entry to the ultimate league fights. They can be obtained from boxes and purchased for gems and in special offers. Please note that there is a limit to the number of tickets you can hold at one time, and any tickets gained over the limit will simply be burned. You can see the limit as the last number, for example, 4/5 means that you have 4 out of 5 allowed tickets. 

Now how to use them: go to your ultimate garage and tap "Join" on the Ultimate league tab in the bottom right corner. Each ticket allows you to enter the series of fights, you get 20 opponents to go through, but you have the chance to leave after each fight (with the accumulated score if you won, with the chance to save the rating for gems if you lost).