How does this old fan regulator circuit work?
The ceiling fan has a split-phase, capacitor-run motor with variable-voltage speed control.
Before thyristor control took over, series voltage-dropping resistors and, rarely, auto transformers were used for speed control.
This regulator comprises an autotransformer with taps and a tap selector switch to select the voltage and hence the speed.

The energy loss in the series voltage-dropping resistor type regulator is avoided with the autotransformer.