Once you have claimed your Star Box, it goes into a special Star Box storage you can access by tapping the blue satchel in the bottom of the Island screen and then the Star Box icon.

The small number on the icon indicates how many unopened boxes you currently have in your inventory. 

 Once you have opened the Star Box inventory, tap on the green button to open a box to see what is inside!

Regular Star Boxes: Each regular Star Box contains three Cards, one of which is at least a "rare" Card. With some luck, you might even find a "special" Card.

To flip the Cards, tap on them one by one or tap the Reveal button to turn all three Cards at once and see what your gifts are. The gifts are credited to your balances automatically when the cards flip over.

5-Star Boxes: Each special 5-Star Box is rarer and more valuable. 5-Star Boxes also contain three Cards, but one is at least a "special" rarity Card with an extra valuable reward.

You can collect the Cards one at a time, or tap the Reveal button to turn all three Cards at once.
Good luck!