Big Fight allows you to be instantly promoted to the next stage of the Championship by winning a certain number of battles against all other players in your group without changing your car.
You can enter Big Fight by tapping the respective button on the stage screen. The number of fights you need to win to get instantly promoted depends on the stage you're currently on. Beginning with stage 4, you will need to win against all of the opponents in the stage to be instantly promoted.
Big Fight button only appears after completing the tutorial.