How does the Villard circuit double voltage?
It doesn't actually change the peak-to-peak voltage of the AC waveform.  What it does do is impose a DC offset onto that AC waveform.

So, a wave that is say +/- 12V becomes a 0-24V waveform (less a little bit for the diode voltage drop).

The capacitor is charged up when the waveform goes negative (through the diode), and releases its charge when the waveform goes positive.

Here is a link to the Falstad Circuit Simulator with a Villard circuit.  You can see how the waveform stays the same but is shifted upwards.