On iOS your progress is synced to your GameCenter account. If you open the game on another device with the same account, your up to date character and progress will be there.

The game signs in with GameCenter and syncs the player in the cloud to that account, if you weren't able to sign in there is a warning message that says "Guest accounts cannot be restored if the game is uninstalled. Please link your account in the game settings."

Tap your players name in the Main Menu screen to access Settings and look for the "Link Account" button to link your account with GameCenter.