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 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.