a potentiometer is used to vary the pulse length. What I'm having trouble understanding is why the right pin of the pot is connected back to the wiper and not to ground, I built the circuit on a breadboard and indeed when I connect it to ground the circuit does not function properly.
The circuit needs a variable resistance there (AKA a rheostat). It does not need a potentiometer, but potentiometers are what you can buy in the store. You use a potentiometer as a rheostat by connecting it as shown, or by leaving one leg unconnected (it's usually better to connect it as shown, in case the wiper loses contact with the resistive element).
Note that potentiometers are not always connected between power and ground, or power and minus power, etc. -- they show up connected every which way; from power (or some reference, or input) to ground is the most common, but by no means the only way.