Coins are used for hints. If you're stuck in a puzzle, tap the hint icons at the bottom of the screen to spend coins and add help to the puzzle. 

You can also spend coins to start a puzzle on an easier difficulty. After selecting a puzzle, choose Quick or Super Quick to make the puzzle quicker to play.

There are various methods to earn coins, found in the in-game store. Tap the coin meter at the top of the screen to open this.