I understand all of that but what I don't understand is how I can take that input and close a circuit.
Your software program would read that input and using a digital output pin perform a digitalWrite command turning a output pin on or HIGH (+5vdc). This digital output voltage then needs to wire to external components (resistor/transistor) that would then drive a simple relay, and the relay's two contact output pins would close (conduct) completing your external circuit.
Lefty