What is the purpose of R3 on this circuit?

R3 provides enough current to thr Zener diode to ensure that it is in a stable condition past the reverse breakdown knee voltage. Without it I suspect that you will find the regulation is not so good over a range of input voltages.

Try running the simulation with a DC sweep on the input and graph that and the output.

I simulated it but even without the R3, the circuit still works.

The circuit will still work even without the Zener diode (with grounded emitter). It is slangly called a "rubber diode", or more scientifically a "VBE multiplier", since it functions as an adjustable "Zener diode". In this case, the base-emitter voltage threshold will serve as an input (relatively) constant voltage VBE.

In this topology, a voltage-type negative feedback is introduced by connecting a voltage divider (R1-R2) between the output (the collector) and the input (the base). In the simplest version (used as an input part of the simple BJT "current mirror"), the collector is directly connected to the base. In this case, the output voltage is simply VBE.