How does an ATX power supply turn off?
The motherboard doesn't turn the power supply on and off. It controls it between a low power standby, where only the 5v_STANDBY line supplies power, and full output.

What controls these two states is the (perhaps mis-named) PS_ON# line. Logic on the motherboard, powered by the 5v_STANDBY line, pulls PS_ON# low permanently to enable the power supply main output, and releases it high to disable the main output.