How does this non-isolated circuit create 12VDC from mains supply?
Your simulation circuit is not connected properly. Go back and look at the original circuit and note that the cathodes of both of the 1N4002's are connected together. This common connection forms the +DC rail of the "off the mains" power supply.



Note also that the zener diodes are not working so much as a reference but as a +12V clamp to limit how high the +DC rail can rise in voltage. This in fact is what "regulates" the DC voltage (If you can in fact call it regulation when in fact all it does is "limit" the voltage. With the original circuit as shown the load on the +DC rail needs to be kept under about 20mA to allow the rail to get up to 12VDC level clamped by the zener diodes. A greater load results in a lower DC voltage due to the current limiting through the 0.68uF capacitor and the 68 ohm resistor. Also interesting is that with the 470uF filter capacitor it takes almost 1 second to get up to the 12V level with a 24mA load.