If you hatch an egg and receive enough duplicates of an existing animal. The meter below the animal will begin to fill.  The meter below the animal will show how many duplicates are needed to level up the animal. The animal will level up when the meter is filled.  

Leveling up means you will receive additional rewards for completing activities.

Some levels may change your animal's coat. To change your animal’s coat, go to your Wildlife Collection (tap the paw icon on the bottom of the game’s homescreen), then tap your animal. This will allow you to rotate through all your currently available coats.