A standard flashbulb firing circuit would connect a battery, a capacitor, a resistor, the flashbulb and the shutter contacts all in series. When the bulb is inserted it completes the circuit and the capacitor starts to charge. The resistor is included to limit the current flow so that the bulb does not fire as soon as you plug it in.

The shutter contacts are wired such that when they are closed, the capacitor is connected directly across the bulb therefore dumping its full charge into the bulb.

It is done this way rather than simply connecting a battery via the contacts as a battery has a relatively high internal resistance which would give a bit of variance on the point in time in which the bulb fired. The capacitor's lower internal resistance allows it to deliver a short high current pulse which will set off the bulb more consistantly.