As DAveman said, the heat is used to expand the air, which causes the piston to move down the bore. etc etc.

As for hot exhaust temp...the hotter it is the less dense it is..

the exhaust removes a MASS of gas. the reason hotter is better is : because the gas has a higher velocity, it helps draw out all the gas from the engine. It does this by creating a lower pressure region in the exhaust (compared to combustion chamber) so the exhaust gassses want to leave.

Another reason hotter exhaust gasses may be better is because : beacuse the gas is hotter it has more energy thus it is easier to make it leave the engine.

But all of this has nothing to do with thermal efficiency. Thermal eff is a part of engine design but, in the simple context I understand it in, it's not the holly grail of engine power, it's more likely, that it is closely related to fuel efficiency.

Hotter is probably better, I would like to know up to what point though???