How funny have you been in front of the cameras lately?
Favor is the stat that measures how The Watcher feels about your performance on the island. The higher the number, the more entertaining you've been, and the more likely she is to give you the time of day when you inevitably wipe out and get dragged off to Failboat.
Favor can be used to purchase shots of adrenaline1) on the Failboat, allowing you to fight in cage matches longer. Alternatively you can trade 100 favor in for a return to the island, saving you from having to wait for the next New Day.
In previous seasons of the show, the Watcher had a bit of a sense of humor and provided much weirder services.