It does not eliminate arcing across the switch contacts; but it reduces.
Think in terms of the total energy: when the current collapses, a part of the energy may be stored into the capacitor (a significant part may be sent to the secondary coils). The stored energy in the capacitor will be fed into the coil when the contact is made again.
The problem is that when the contact is broken, the current must change very rapidly and the induced voltage must be very large (a singular process). So you need a very high voltage capacitor.
The ideal capacitor value will be determined by the LC time const matching the spring time const (for the make and break switch). Tricky to figure out but not impossible.