The reed switch is a little switch that turns on when it's near a magnet and shuts off when the magnetic field goes away:
http://en.wikipedia.org/wiki/Reed_switch
You would ground out one side of the switch, and run the other side to the relay. Then you would mount it underneith the little pocket in the dash, and you would glue or stick a little magnet on the door of the pocket.
When the door is closed, the magnet would be right above the switch and would cause it to "close" which would start sending current to the relay, which would shut off the lights. When the door opens, the magnet would be away from the reed switch, which would make it "open", which would stop sending current to the relay, which would turn the lights on.