Trophies are awarded at the end of each Battle and measure your overall success in the game. The number of Trophies you win after a race is based on your position in the race, as well as the number of Trophies your opponent's have.
For example, if you beat players who have more Trophies than you, you'll win more Trophies than if you beat players with fewer Trophies than you. Conversely, if you lose to a player with more Trophies than you, you'll lose fewer Trophies than if you lose to a player with fewer Trophies than you.
As well as determining your Tier and the rewards you receive, Trophies are also used for matchmaking. When you start a new Battle, the game will attempt to match you against players who have a similar number of Trophies.