Why does a resistor reduce voltage if V=IR?
According to Ohm's law, resistance varies directly with voltage


You should read this the other way. Voltage varies directly with current. "R" is the constant of proportionality telling how much it varies.


  If I add in a resistor to a circuit, the voltage decreases.


If you have a resistor in a circuit, with a current flowing through it, there will be a voltage dropped across the resistor (as given by Ohm's law). If the resistor is in series with some other element, and they together are powered by a constant voltage source, then the voltage dropped across the resistor means there's less voltage available for the other circuit element. It doesn't mean that the voltage of the source decreased.