How does this MOSFET based motor driver designed work
I'm not sure how the MOSFET operates to boost the PWM signal


The MOSFET is acting like an on-off switch changing from on to off as quickly as you switch your PWM waveform. So, if the PWM is 50% duty cycle then the average voltage across the motor is 50% of 4.8 volts = 2.4 volts. If the PWM was permanently a "1" then the motor voltage would be 4.8 volts.


  I placed the capacitor in parallel to block noise


That doesn't work too well as a noise reducer because that capacitor is being switched on and off across the 4.8 volt supply at PWM speed. This means that large current pulses will flow through the capacitor and from the power supply and generate EMI. So, rather than acting as a noise reducer, it acts as a noise source. If you really want to reduce noise properly then there should be an inductor between the MOSFET drain and the circuit that connects to it. Then you would need to move the diode back to the MOSFET drain to prevent inductor back-emfs destroying the MOSFET.


  what the purpose of the resistors is


The resistor from gate to source (usually around 10 kohm) ensures that the MOSFET turns off when the circuit is not connected to your Arduino. The series resistor (usually around 100 ohm) is to reduce stress on the Arduino output pin that drives the large gate-source capacitance of the MOSFET.


  Any help is greatly appreciated


Noted!