Why does this zener diode fix the voltage to the LEDs?
The description in your link is totally bogus.

Assuming the switch is in the positive lead, the LEDs are shown backwards - the cathodes should be towards the negative supply lead, and there should be one resistor for each LED. (Sharing one resistor between two paralleled LEDs is a Bad Idea, particularly if the LEDs are different colours.) There will be 4.7 volts dropped across the Zener diode, but that does nothing to stabilize the voltage across the LEDs, or otherwise compensate for varying supply voltages - it just makes the voltage across the LEDs and their resistors about 4.7 volts less than the supply voltage.