I was watching this video and he was explaining that this part of the circuit was to smooth out the voltage to make it stable enough to drive the MOSFETs. Just looking at it, I see it's just an RC low pass filter, but I'm not quite sure why the Zener diode is there. Could someone give a bit of light to what its purpose is?
This circuit is not generally looked at as a low pass filter but as a regulator with bypassing. The Zener and resistor are used as a shunt voltage regulator. The capacitor is to bypass the regulated voltage, this gives the regulated voltage a low impedance to ground for AC signals.
It's also usually drawn with the voltage source to the left, then the resistor, Zener and capacitor to the right. In schematics we try to have power and signals go from left to right, higher voltages at the top, ground at the bottom.
As far a depictions go, I'd draw the circuit as follows:
simulate this circuit – Schematic created using CircuitLab
This makes it perhaps more obvious that R1 is dropping down voltage, and that D1 is a 6.2V shunt regulator. To get D1 performing well, it needs say 5mA flowing through it (at least), and in this case R1 is chosen to provide just such a current with 10V input. Of course, if the input voltage varies over a wide range, R1 will degrade the performance of the diode, and a current source will need to be used instead to get the best performance the Zener can offer. Of course in your application it doesn't really matter I think.
make it stable enough to drive the MOSFETs
"Drive" as in use it for gate voltage? I wouldn't worry about that. 10V will work fine as a gate voltage for power mosfets, as will 15V, and perhaps even 8-9V, although that depends on a mosfet. The gate voltage doesn't have to be very accurate. 10V-20V will work well for many, many mosfets.
Sure, with higher gate voltage the gate charge will be higher, but I doubt that was a big concern in the design.
Gate voltage for switching applications needs to be high enough to turn the mosfet fully on, and must not exceed the maximum gate-source voltage. That's usually quite a wide range.
