The purpose of the Krebs cycle is to generate nadh in order to power the electron transport chain. Up until this point, respiration has produced relatively small amounts of ATP. Now we finally get the big payoff. Hydrogen ions from NADH are transferred across from the interior of the mitochondria to the external matrix, generating a potential difference across the internal membrane. Why is this good? Potential difference means electrical current. We get a flow of electrons across the internal membrane which are used to power ATP synthase. Eventually the electrons become reunited with hydrogen ions and we use oxygen as an acceptor (the reason for the requirement of oxygen in aerobic respiration) which produces water which is nice and easy to deal with.

The increased ability to produce energy that mitochondria provide is probably one of the reasons for us eukaryotes being up here and prokaryotes never getting round to discovering fire, inventing the wheel, forming civilisation, that sort of thing.