Why connect the source pin on a mosfet to the pull down resistor?
I am not sure if I understood right, but I will try to answer:


Why there is ground line going back to the Arduino board?


Because all grounds in DC circuit must be connected together. Simple as that.


Why is there a pull-down resistor on MOSFET?


The MOSFET you are using is N-channel MOSFET. It is "activated" or "open" where voltage on the gate is larger than "Gate threshold voltage", the value you will find in MOSFET datasheet. You want your MOSFET to be off in all times unless you apply voltage to the gate. 

In this case it is actually a pull-down resistor from gate to the ground(hence the pull-down resistor). It is coincidence that in N-channel MOSFET you connect source to the ground so it looks like there is a MOSFET on source line, but it is actually just a pull-down resistor on gate to keep MOSFET off.