A voltage is a difference
in electric potential between two points. Key word emphasized. Electric potential is the potential energy per unit charge
(measured in volts = joules per coulomb). So if you imagine a unit test charge of +1 C moving from an area of low electric potential (like the battery negative terminal) to an area of high electric potential (like the +terminal) it will gain an amount of potential energy in joules that is numerically equal to the voltage (potential difference) in volts.
Of course, in reality what happens is that negative charges move from an area of high electric potential to low, and in so doing, they gain potential energy (because they are negative, and so the electric field affects them in the opposite way).
The increase in electric potential energy occurs because the charge, when going through the battery, moves in the opposite direction of the electric force (from the field), and so the electric force does negative work on the charge (just like when a ball is thrown up, gravity does negative work on it and it gains gravitational potential energy).
The answer is that electrons everywhere in the circuit start flowing all at the same time. So charges that are already in the wires start moving. (A conductor contains a whole bunch of free electrons.) The reason why this happens even though the source of the potential difference is at the battery is because if charges started flowing out of the battery, but the charges ahead of them in the wire didn't
move as well, then there would be a pile up of charges. However, like charges repel each other. In other words, the pile up of charges would lead to an electric field that would tend to smooth out the pile up of charges. So, everywhere in the circuit, locally, there is an electric field that causes a steady net flow of charge.