Experience measures your progress on the game. By finishing any battle, winning or losing, you will obtain some experience.
The amount of experience gained on battles depends on your performance. Winning battles will reward you more experience. Being the "Star Player" will also award you with extra experience.
Every time you level up you will get 20 Tokens as a reward.
The experience required to level up to Level 2 is only 40 Experience Points. Every level requires 10 extra Experience Points, making it harder to progress as you level up.
Your Experience Level doesn't affect the game at all. The matchmaking is based only on the amount of Trophies the Brawler you are using has.