They use the spin of the electrons. If you apply a magnetic field to the
YIG,the spinning electrons precess at a rate proportional to the
magnetic field. This rate happens to fall in the microwave region and a
typical range is from 8GHz to 18GHz.
The sphere is coupled to the oscillator circuitry by various means - a
loop of wire for example - and the spinning electrons look like a
resonator to the circuitry.
Alan