How does this drop the voltage down without a resistor?

[Beamin]

This goes from live 220v to 24 or 17 volts. How does it do this without a resistor or regulator?

[MikeK]

What do you mean?...The zener is setting the voltage.

[themadhippy]

the 330nf capacitor is  pretending to be a  resistor,the magic phrase to search for is capacitive dropper

[TimFox]

Since the reactance of the capacitance is much higher than the load resistance (Zener or relay), it limits the current nicely from the high voltage source into the load.  A good capacitor will not dissipate power and heat up, while a 10 k resistor would dissipate about 5 W with 200 V across it.  That is approximately the reactance of 330 nF at 50 Hz.