Puzzle pieces are part of the puzzle events. There is a chance to earn a puzzle piece from normal and hard levels, after earning a crown on a crown level, or by receiving a piece sent from a friend. If you complete all 15 levels in the Mighty League, you are rewarded with the opportunity to pick a puzzle piece that you need!
You view your puzzle pieces and add them to your puzzle within the puzzle event screen. On the main level screen, on the right-hand side, you'll see an icon that looks like a puzzle piece.
Tapping on that brings up the puzzle event. Here you can see which pieces you've collected (you can scroll left or right through the pieces to see all nine), and tapping on a piece you've collected gives you the option to add it to your puzzle, or send it to a friend.
Once you’ve added a puzzle piece to your puzzle, you can always remove it again. Just tap on the piece in your puzzle, and it will be added back into your collection.
Once you've completed the puzzle, you can claim rewards! You can even complete the puzzle multiple times per event and receive rewards each time. You can see what the rewards will be by tapping the “View” button. Each puzzle event runs for one month, at the end of which all puzzle pieces that have not been redeemed will be lost, so be sure to complete your puzzle and claim your rewards before the event ends!