Puzzle pieces are part of the puzzle events. On your main journey map, on the right side of the screen, 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, and tapping on a piece you've collected gives you the option to add it to your puzzle, or send it to a friend. It's possible to get multiples of the same piece, so you'll probably end up with extras to send!
You are able to get puzzle pieces randomly in any level, after passing a hard level, after earning a crown on a crown level, or by receiving a piece sent from a friend.
Once you've completed the puzzle, you will be able to receive rewards, which you can see 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 claim your rewards before the event ends!