How does this MOSFET/Op-Amp voltage regulator circuit actually work?
First note that the feedback from the output is connected to the inverting input of the op-amp. That means were dealing with negative feedback and the op-amp output will stabilise when the non-inverting input (1) and inverting input (2) are at the same voltage.

At power-up Vout is zero as Vin begins to rise.
The voltage at (1) will rise to Vref.
Since the feedback signal (2) is close to zero and the reference voltage (1) &gt; 0 the op-amp output (3) will start to swing towards the positive supply.
The transistor is wired as an emitter follower so the voltage at (4) will be V4 = V3 - 0.7 V due to the base-emitter voltage drop.
The voltage at (5) will rise and this feeds back to (2).
As V2 gets very close to V1 the output of the op-amp will start to fall until V2 = V1. At this point V5 = Vref.

So if V5 = Vref then \$ V_{\text{out}} = \frac {R_1 + R_2}{R_2}V_{\text{ref}} \$.
You've changed your schematic to a MOSFET since I started my answer but the principle is the same.

From the comments:

How does the Zener diode fit into this?

Note that the Zener diode is pointing towards the positive supply so that it is reverse biased. A regular diode will not pass any current in reverse mode1 unless you exceed the reverse breakdown voltage at which point you have probably destroyed it.
Zener diodes are designed to breakdown in a controlled manner at a specific voltage. All that is required is to feed it a voltage higher than it's breakdown voltage with some form of current limiting, R3 in this case. The voltage at (1) will be reasonably steady for a wide range of voltages Vin &gt; Vref. In practice you'll need a little &quot;headroom&quot; of a couple of volts at least.
1 Not quite true. There is a little leakage current and you can get its value from the datasheet.


So ... what heavy lifting does MOSFET do?

Op-amps typically have an output capability of a few tens of mA max. The addition of a BJT or MOSFET increases the output current capability. It's like power steering; the op-amp is the little old lady at the wheel and the transistor is the hydraulic amplifier moving the steering rack.