When the engine is running, the purge solenoid opens to allow the fuel vapors to be drawn into the engine and burned. When the engine is off, the purge solenoid closes to prevent the fuel vapors from escaping into the atmosphere.
If the purge solenoid is stuck open, it will allow fuel vapors to escape into the engine, which can cause the engine to stall or run poorly. If the purge solenoid is stuck closed, it will prevent the fuel vapors from being drawn into the engine, which can cause the engine to run rich and stall. In either case, a bad purge solenoid can prevent the car from starting.