From the Super Mario Wiki
It involves the player attempting to match the touch screen picture with the top screen picture. The screens are made of many panels that have Super Mushrooms on the front and Cape Feathers on the back. To match the two screens, the player must touch a panel with their stylus to make it turn around. However, flipping around a panel makes the surrounding panels flip around as well, adding a challenge. There is a game called Puzzle Panel that is the same game, except with less panels. Unlike Puzzle Panic, Puzzle Panel appears in New Super Mario Bros. as well. If the panels are flipped correctly, similar to Puzzel Panel, the two Yoshi statues on the left and right blasts off in the air and the player advances to the next level.
 In-Game Instructions
This minigame takes the Puzzle Panel to a whole new level. Put on your thinking cap. You're going to need it.