This feature becomes available at Level 19. It is started by finding a flyer during an excavation mission. After returning to the Park all the pens will have a housing built inside them. Tap each housing to begin dressing up the dinosaurs from that pen.
Select the dinosaur you want to dress up by tapping on the Left/Right arrows. Next, select the item category from the top shelf, then select the item to purchase.
Dinosaurs carry their items into battle, so your opponents will see them.