A neutral safety switch allows you to start an automatic transmissioned car only in PARK or REVERSE. This way the car can only move when you put it in gear.
That is to make sure that the starter only operates when an automatic transmission is in park or neutral. Or USA cars with manual transmissions also have a switch to make sure that the clutch is pushed in and not engaged while starting the engine. Basically it is to make sure that the vehicle does not start moving due to starting the engine.
It is used in automatic transmissions to prevent the vehicle from starting in gear and possibly causing an accident.
