The Star Stone Bag is the user interface for working with Sockets and Star Stones.
Here you can add Sockets to equipment (and remove it) and then add Star Stones to the Sockets. It's also the place to fuse and store Star Stones.
See the pages for Sockets and Star Stones for details about them.
Access your Star Stone Bag by clicking the right-side shortcut, then right-click the Star Stone in the inventory, and then select Game Info - Star Stone Bag.