How does the computer power supply (auxiliary circuit) work?

I don't know what you mean by "auxiliary circuit". The PC power supply outputs multiple channels, one of which is 5V standby and always outputs, as long as the power supply is supplied. The output of other channels depends on whether the computer is turned on or off. These channels are not output when closed, and only output when opened.

The on/off button of the computer is powered by that 5V (standby), which provides power input signal to control the output of other channels.