Take, a battery. It has many negative charges gathered at one pole. They repel each other strongly, since they all produce a negative electric field.
If you connect a wire to this pole, then suddenly the charges feel a neighbour location with less repulsion. Imagine being the charge at the edge of the charge bundle, feeling a lot of electric repulsion force from the bundle, and now suddenly a path opens up next to you with less repulsion. This charge will immediately move to this path.
Where that charge came from is now an empty space with less repulsion for all the other charges. So, another charge will immediately take over this spot. This means that the first charge that moved now again feels a repulsion from behind, and it thus moves further ahead.
All other charges feel the same less repulsion on this wire-path. They all want to move there. They will thus "queue" up to move along this path. The charge in front is moving due to the push from the electric field of the charge behind it. And that charge moves forward due to the electric field of the charge behind that one.
Eventually, the entire circuit is filled up with charges.
Note: This is a simplified picture, since there area already charge in balance filling up the wire. The repulsion is still less along this path, so negative charges form the battery pole will indeed push on the charges that are already along the wire. They will then in turn push their neighbour charges - and this is propagation of the "push" happens at the speed of light.
The repulsion force is an electric force. A larger electric force is what we associate with a larger electric potential energy. On a per-charge basis we call it simply electric potential. Whenever you have two points with not the same repulsion against a charge, then that means two points at different electric potentials.
A difference in potential is what we call voltage. So, there we have it: Voltage is just a word for a difference in electric repulsion; a word for a charge's tendency or willingness to move to that other point. The voltage thus describes the "pressure" that makes charges flow throughout any circuit.