The original Statement is valid.
"the potentiometer controls the voltage to the Arduino"
In the tutorial, you are hooking the resistive element of the potentiometer across ground and VCC rails.
The wiper is then connected to your input.
You sir, now have a voltage divider. Current will remain a constant, (Whatever the value of the pot is) and it is the ratio of voltage relative from each rail to the wiper that changes.
http://arduino.cc/en/Tutorial/AnalogInput
By turning the shaft of the potentiometer, we change the amount of resistence on either side of the wiper which is connected to the center pin of the potentiometer. This changes the relative "closeness" of that pin to 5 volts and ground, giving us a different analog input. When the shaft is turned all the way in one direction, there are 0 volts going to the pin, and we read 0. When the shaft is turned all the way in the other direction, there are 5 volts going to the pin and we read 1023. In between, analogRead() returns a number between 0 and 1023 that is proportional to the amount of voltage being applied to the pin.