The fuel cut-off stop solenoid valve opens and closes the fuel passage leading to the suction port of the pump (The valve remains open while the ignition switch is on.)

When the battery voltage is applied to the solenoid the valve compresses the spring to be drawn upward. As a result, the suction port is opened and the fuel is supplied.
When the current to the solenoid,is cut off, the valve to retracts downwards by the spring. As the valve closes the suction port, the fuel supply to the cylinder will stop, causing the engine to stop.
If your solenoid has failed it can prevent the engine from running or prevent it from switching off.