The Prince uses the Gauntlet for multiple purposes. In combat, he uses it to throw enemies into the air, break certain shrouds around enemies, and chain together combos for more damaging attacks.
The Gauntlet is also used to gripfall by digging the claws into the wall to slow the prince's descent, as well as helping in wall running.
When the Prince is asked by Elika where he acquired the Gauntlet, he simply replies that he found it. Other conversation between the two reveal the Prince is a tomb raider, suggesting he may have stolen it while raiding.