I know that the purpose of salt in the ice is to set the ice cream, but what makes the salt do that, especially since salt is known to melt ice?

Salt does melt ice, but only to an extent. The main action of salt on roads (beside damaging them) is lowering the freezing point of water. When the salt hits the top layer of ice, it is absorbed by the water at the top of the ice. (You may not see the water, but it is there.) This changes the freezing point, lowering it, and then the unfrozen salty water works on the rest of the ice. The water will not re-freeze because of the salinity.
When you are making ice cream, you want the mixture to freeze. But it can't freeze if the temperature immediately around the mixture is at 32* or above. Adding the salt to the mixture lowers the freezing/melting point, which allows the liquid to get colder and freeze the mixture. The heat in the mixture is drawn out into the saline solution. The mixture then gets colder and freezes.

Thank you DG!!!