How in the respiratory chain do electrons from FADH2 and NADH2 passing through cytochromes liberate energy for the ATP synthesis? What is this ATP synthesis called?

FADH2 and NADH2 oxidate into FAD and NAD and liberate hydrogen ions and highly energized electrons in the beginning of the respiratory chain.

The energy lost by electrons that pass by the cytochromes is used to pump protons (hydrogen ions) out of the inner mitochondrial membrane (to the region among the inner and the outer membranes of the mitochondrion). Hydrogen concentration gradient among the inner and the outer spaces delimited by the inner membrane forces protons (hydrogen ions) to return to the mitochondrial matrix (the region inside the inner membrane) though that return is only possible if hydrogen ions pass by an enzymatic complex called ATP synthetase embedded in the inner membrane. In that passage the ATP synthetase phosphorylates ADP and then ATP molecules are produced.

Hydrogen liberated in the mitochondrion then combines with oxygen to form water. As a reaction that depends on oxygen this type of ATP synthesis is known as oxidative phosphorylation.