Explain why the collector voltage is approximately zero when a transistor has a collector-emitter short?

The collector voltage is not necessarily approximately zero when a transistor has a collector-emitter short. It depends on whether or not there is an emitter resistor.

A typical collector-emitter circuit has two resistors, one in the collector and one in the emitter. One or both of them might be zero, i.e. not present, depending on design requirements. The collector-emitter junction represents a third resistor, the value of which is dependent on base-emitter vs collector-emitter current ratiOS and hFe.

If the collector-emitter junction is shorted, then this circuit degrades to a simple voltage divider, or single resistor, and the collector-emitter voltage differential will be approximately zero. Simply calculate the voltage based on the one or two resistances.

Results could be different than calculated, if the resistors are small in camparision to the shorted impedance, and it could be different depending on the base to emitter or collector relationship in that fault state, though the latter case is usually negligible due to the relatively high resistances of the base bias circuit.