Why Power dissipated in the active device is the product of the voltage across it and the current through it. A switching amplifier (Class D, E or F) drives the transistor hard between full conduction (saturation, so very low voltage across it) and cutoff (essentially no current through it), and in both states that V x I product is nearly zero. Only the brief transitions between states waste power, so efficiencies of 90 percent or more are possible, versus roughly 25 to 66 percent for linear Class A, AB or B stages where the device sits at intermediate voltage and current.
Watch out High operating voltage has nothing to do with it; a linear amplifier at any supply voltage still dissipates heat because the device spends its time partly on. The harmonic content comment is backwards too, since switching amplifiers generate lots of harmonics and need heavy output filtering.
Efficiency comes from P = V x I being near zero: full on means no volts, full off means no amps.