Why does the voltage of a zener diode match the voltage at Vout?
Zener diode passes a lot of current (has low resistance) if the voltage is above Vz. So, connected in a circuit like in the question:


If the voltage at the output was lower than Vz, the diode would not conduct and (if there was o other load, the voltage would rise.
If the voltage at the output was higher than Vz, the diode would have low resistance and short out the output, which means that the voltage would drop (because of the resistor in series).


So, the circuit reaches an equilibrium of Vout=Vz. If the voltage tries to go lower (a load is connected), the diode conducts less and the voltage rises back up to Vz. If the voltage tries to go higher (aload was disconnected) then the diode would conduct more and drop the voltage down to Vz.