What is the purpose of a capacitor to ground?

A capacitor's complex impedance is given by

$$ \frac{1}{j \omega C} \Omega $$

where \$\omega\$ is the frequency, and \$C\$ the capacitance. For a low-frequency signal, you can see that the impedance will be relatively high. Conversely, you can see that for a high frequency signal, the impedance will be quite low.

A high-frequency signal will see the capacitor connected to ground, and travel through it, since it is a low impedance path, but a low frequency signal will not be affected by it. The capacitors to ground form a low-pass filter for the lines they're connected to, as they remove high-frequency signals from the line by giving those signals a low-impedance path to GND.

See this question.

I expect C1, C2 and C3 in your diagram are filtering capacitors. They filter unwanted high frequencies from power line. Their impedance is low for high frequency signal and high for low frequency signal. This results in acting like a short circuit for high frequency signals. All these capacitors are in dangerous places - in the case of their failure. Because of this, special X and Y capacitors are used in these places. I expect your C1 is X2 rated, while C2 and C3 is Y2 rated.

You can find more information why this is used if you search on Google for Y2 capacitors.

I think you should draw L1 and L2 differently, because it is a common mode choke on a single core.