When you tap on the trophy icon in the upper right corner, you get to the league screen.
On the left side, you see all available leagues and the one you are currently in. The bigger part of the screen makes up the leaderboard itself on which you can see your own points and your competitors. The location of all players is also displayed.
Tap on a player on the leaderboard to go to his village. This can help you get new and better layout ideas.
The longer you survive, the more points you get. Once you reach a certain score, you will be instantly promoted. Depending on which league you are in, you get certain weekly rewards. The higher the league, the better your weekly rewards.
Once a season ends, the rewards you can win are changed and the top players will be reset to 8100 trophies.
Find out below how many points you need to reach each league:
Wood League: 0 - 999 points
Iron League: 1000 - 1900 points
Bronze League: 2000 - 3999 points
Silver League: 4000 - 7999 points
Gold League: 8000 - 8999 points
Diamond League: 9000 - 9499 points
Survival League: 9500+