Two analogies:
1. The wires are like hoses full of water. As soon as you turn on the water, water is pushed out the end of the hose.
2. Although the electrons only move at 1mm/s, the electomagnetic pulse travels through the circuit at near speed of light (varies btw .1c to .9c). It is this pulse that provides the pressure to push the electrons out the end of the wire into the light