How does a supersonic electric jet airplane work?

They don’t work.

There are no supersonic electric airplanes.

But you are correct in the sense that “turbines act as super propellers” in fossil-fuel jet engines.

Wikipedia – Turbojet

Wikipedia – Turbofan

An electric motor would have to drive the turbofan fast enough to provide supersonic speeds.

I was under the mistaken assumption that heat itself was creating the thrust in the conventional jet engines, and that air was a part of the combustion process to create the burn that provided the thrust.

@Call_Me_Jay points to good info about turbofans. Only a fraction of thrust on a conventional airliner is directly from combustion producing high-velocity hot gasses out the back. That’s the “jet” part… the jet engine also drives a shaft that turns big (turbo)fan blades in the fron of the engine that accelerate air around the jet engine and produce most of the thrust.
These fan blades are not suitable for supersonic speeds, most of what I see about that mentions massive vibrations and instabilities as airflow approaches the speed of sound.

I have not seen any proposals for supersonic electric propulsion.

I was under the mistaken assumption that heat itself was creating the thrust in the conventional jet engines, and that air was a part of the combustion process to create the burn that provided the thrust.

I think that is accurate, not a mistake. All jets use air.

Turbojets are rocket-like, compressing lots of air to ignite lots of fuel, expelling a mass hot gas.

Turbofans are similar, but turn more of the energy to turning fans to expel masses of air outside the combustion.

@Call_Me_Jay I’m not sure anymore.

Does the jet engine rocket the plane forward like I thought out of the back of the engine, or does something happen around the outside of the engine?

@Ltryptophan Both. In a modern turbofan, the jet engine does “rocket” the plane forward with hot combustion gasses but more of the power is used to drive a shaft that turns the big fan blades. It’s just much more efficient to turn a fan blade at lower speed than to put all the air through combustion (not necessary) and high compression of the jet.