Leveling up will allow you to get characters that will help you complete the collections and get different rewards. To level up you'll have to earn crowns, which you'll be able to get in classic games, tower duels, daily challenges and by completing missions.
On the home screen, you will be able to see the level you are currently in on the upper left corner, represented by a blue star.
In addition, on the same screen, you will be able to see how many crowns you need to reach the next level, in the "Special Chest" bar. Keep in mind that, as you level up, the number of crowns you need to continue advancing will increase.