The compressor pressurizes an air tank. When the pressure in the tank is nice and high, use the compressed air to turn a turbine connected to the crankshaft of the engine.

This technology of starting a diesel engine using a turbine driven by compressed air was used in Russian T-34 tank during the WWII. While Germans could not start the tanks in the cold of winter 1941 from the frozen batteries the Russians were using compressed air (hand-crank) to start T-34s just fine.

You can also directly feed the compressed air into a cylinder (or even the intake manifold!) to force the engine to turn. No extra turbine required, though the plumbing might get a little odd. [https://en.m.wikipedia.org/wiki/Air-start_system]

That tends to be for very large engines, where the extra plumbing isn’t a problem.

It doesn't. It's a power storage device to allow muscle power to produce a higher peak output.