Why do people wake up sweaty after a nightmare?

Answer 1

Because the brain cannot differentiate between a picture projected by the eyes or the mind's eye.
During a nightmare the brain still receives signals that cause it to pump adrenaline into the bloodstream which results in hyperventilation which needs to be cooled by sweating.

Answer 2

Because they probably had to run their asses off trying to get away from some big hairy spider-snake mutant or their mother-in-law!