how does this power supply work?
D1 creates a fixed voltage by acting as a shunt regulator.  R1 supplies enough current thru D1 so that it is run where its voltage is fairly constant over a wide range of current.

That is all you need if you just want a reference voltage, like into one side of a comparator for example.  This reference can only deliver at most the amount of current coming thru R1 minus the minimum D1 needs to stay at its stated voltage.  In other words, just R1 and D1 together make a steady voltage, but with rather low current capability.

Q1 is used in the emitter follower configuration to provide current gain.  This creates a power supply with much more current capability.  Actually the current capability is (gain + 1) higher than just R1 and D1 by themselves.

There is a drawback though.  Q1 will drop some voltage from its base to emitter.  The B-E junction looks like a diode to the circuit, and will have a typical diode drop, usually somewhere in the 600-750 mV range.  The net result is that the output at the emitter of Q1 is a power supply with a voltage around 700 mV less than the zener voltage, and a lot more current capability than just the R1 and D1 "supply".