Here's how I've done it: in fair , warm weather, you spread the tarp out on the ground to protect from damp and sleep on top of it in a bivvy sack or sleeping bag. In colder or foul weather, you use the tarp to make a shelter by casting it over tree branches and anchoring it to the ground. And yes, you could also use it as a blanket, or tie it to a tree.
The tarp is a basic survival piece. What's weird is that he insisted to MB that he had only the tarp, not a bivvy sack or sleeping bag. That was one of the unbelievable statements that made her suspicious. No one with any of the knowledge he claimed would go out equipped (or rather unequipped) like that.