The Leagues are what define the rewards you receive after battles, and each league level generates a different reward at the end of each battle. It is a weekly ranking that is based on your score, and every week the scores are reseted. Depending on your ranking position you can rise or fall from the current league.
The position in the ranking will depend on the highest scores, so there is no minimum or maximum score required to enter a specific league. If the best score during the current week is 200,000,000 kills, the highest League will include a certain amount of players that managed to achieve a similar score, and the other players will be positioned accordingly in the ranking. If the best score was 50,000,000 kills, the same rule would be applied.
There are currently 5 Leagues, and each League has 5 levels, which will give you a different bonus in your rewards.
Battle rewards are calculated by: Loot + League Bonus
Loot: the coins you collect in the battlefield are what we call loot. These coins have random values, and all coins collected will be summed at the end of each match. So the more coins you collect, the more rewards you will receive.
League Bonus: the league bonus is the base amount of coins delivered to each player according to the Leagues they belong at the moment. This bonus is multiplied by your VIP level when you are a Premium member.