It's similar to manual control over the current stack. You pause execution, you dump your current stack onto the heap, and restore it later. Control over concurrency, similar to out of order execution. I'm describing it conceptually but I assume the implementation can be/was smarter/faster.
I don't think closures have anything to do with it.
There was also some debate about how cancelation of an awaitable function built with those primitives would work. I don't think there was a great answer for that yet.