Why is a capacitor used in a starter?

A capacitor is used to reduce the arcing when the ignition circuit's contacts open and close.

A longer answer

To be answered fully, this question needs to give more information:

Is the question asking about a starter for an aero engine or a ship's engine, or for the engine on a car, truck or other vehicle?

Or is the question asking about the starter for a fluorescent light fitting?

Answer for a fluorescent light fixture's starter

A capacitor is a device which absorbs and stores energy in the form of an electric field.

When we switch on the light, the fluorescent tube initially requires more energy than is required for when it is continually running.

This is because the tube's two filaments - one at each end of the tube - have to be supplied with current to give them enough energy to heat up and glow. Their extra heat makes an arc flash across the tube, between the two filaments.

The arc will then continue running across the tube once the filaments have been disconnected from the power source. * (See Note below.)

After the arc has started across the tube, a switch inside the starter must open to break the supply of current to the filaments.

Without a capacitor, a secondary arc would start across the open contacts of the switch inside the starter and that arc would quickly cause those contacts to burn away. The capacitor absorbs the energy which would otherwise start that secondary arc to glow. That is why a capacitor has to be used in the starter for a fluorescent lamp.

* Note

As part of the lighting fixture there has to be a special inductance coil - called a "choke" - that is wired in series with the fluorescent tube.

The choke's purpose is to limit the amount of current drawn by the arc whilst the tube is lit up. Without the choke the arc would act like a short circuit and would cause the circuit breaker or fuse protecting the lighting circuit to trip or blow in order to cut off the supply of current.