What causes the dashboard light to dim after turning on the headlights?

Because the power for your headlights is pulled from your battery. When the dashboard lights dim after turning on the headlights the dashboard lights aren't getting as much power as they were with the headlights off.

With the motor running or off?

The headlights draw a lot of current, but should not visibly affect other systems.

There could be a problem with a worn out battery, an alternator that isn't charging correctly, or you have a bad earth between the battery and the chassis.

Check the large cable connection between the battery (usually negative these days) and the bodywork, or between the engine block and the bodywork.