A “tenter” is a frame to stretch clothing to dry. A “tenterhook” is the hook that attaches to the clothing to stretch it. Thus, to be “on tenterhooks” means to be stretched to the point of tension.
(I’m assuming “tenter” means “a thing that tents something,” like a verb. In this sense, “being on tent stakes” would mean the same thing, as tent stakes are something which pulls fabric to tension. But, for whatever reason, that didn’t catch on.)
Why I Looked It Up
Someone was waiting for important a phone call which was three hours overdue, and was described as “being on tenterhooks.”