Stacks-GIT develops this theory in (what was at one point) chapter 16 in the chapter on hypercoverings. It is explained in 16.7, although I haven't read it, so I can't vouch for its quality.
Specifically, descent by cohomology is covered in chapter 27, but it relies very much on the machinery developed in chapter 16.
Another useful thing you might want to look at (in French):
Cours 5 develops homotopic descent, which will be very useful for studying lurieLurie and Toen-Vezzosi.