Trophies are awarded at the end of each Battle and measure your overall success in the game. Usually, you'll gain Trophies if you finish first or second place, and lose Trophies if you finish third or fourth. You can compare your Trophies with other players by checking the leaderboards.
Trophies are used for matchmaking. When you start a new Battle, you will be matched against players who have a similar number of Trophies to you.