Why connect the source pin on a mosfet to the pull down resistor? I'm trying to power stuff controlled by an arduino, and I'm looking at this tutorial: http://bildr.org/2012/03/rfp30n06le-arduino/ and it almost all makes sense, send a signal to the gate, but it's like a capacitor or something so provide a path for that signal to go to ground when the arduino isn't supplying current, but when it is supplying current then the source and drain are connected so current can flow from some external source through them... but why even connect that part of the circuit back to the board? I'll paste a pic with a question mark on the wire I'm frowning at:
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.