You asked
What actually happens inside the engine to cause it to function incorrectly at higher temperatures?
Response
One of the many ways that engines fail due to high temperature is the increased diameter of a piston due to the heat. As you have noted heat will make things get bigger. In a situation where an engine seizes due to heat the OD (outside diameter) of the piston increases in size so much that is larger than the ID (inside diameter) of the bore it is traveling.
Upon seizing because of heat, due to very high friction, the piston stops the crankshaft from turning as the motor cannot create enough power to overcome the friction of the seizing piston in the cylinder bore.
Upon cool down of the ICE the piston will shrink and allow the engine to turn over if there is no physical impedance from the actual seizure.
In some cases an engine can be started after a heat related seizure. This does not mean that it is good operational order or that lifespan and durability have not effected it. More than likely rings, piston and cylinder bore have irrevocably damaged and require replacement and machining in order for the ICE to continue to operate normally.
Other Affects
Plain bearings for the crankshaft and connecting rods
Radiator/Coolant hoses can burst due to the increased pressure in the cooling system.
Radiators that are old or poorly constructed can fail due to increased internal pressure of the cooling system
Head gaskets can fail due to the increased temperature of the coolant OR due to warping during cool down or an overzealous operator putting cold water into the radiator before the ICE can cool slowly and evenly
Overhead cam cam holders
Various other components