How is surface charge induced on the surface of circuit wires when connected to a battery? Understanding:
I watched the newest Veritasium electricity video and he said that the electric field inside the wires of the circuit that causes the electric current to flow is created by both the battery and the charge that resides on the surface of the wires.
Confusion:
Before the battery is connected to a circuit, there must be no net charge anywhere inside or on the wire because otherwise there would be a different in electric potential and a current would flow. So that means the battery must induce the surface charge when it is connected which is why a current then flows (given a complete circuit), but how does it do this? How is charge all of a sudden appearing on the surface?
And also, why doesn't a bulb glow brighter closer to the battery, since the electric field is stronger?
Any replies that come forward are much appreciated. Kind regards.
 A: 
Before the battery is connected to a circuit, there must be no net charge anywhere inside or on the wire because otherwise there would be a different in electric potential and a current would flow.

Net charge on the disconnected or connected circuit is not relevant; indeed it can be zero.
The important thing is that there is non-trivial surface charge distribution, at some places positive, at some places negative, and so this distribution as a whole can be non-zero even with net zero charge, before the battery is connected.
Since no current is flowing before the battery is connected, net electric field inside the wire vanishes, so this surface charge distribution is such that it completely cancels out electric field due to the battery and possibly due to other bodies producing electric field.

the battery must induce the surface charge when it is connected which is why a current then flows (given a complete circuit), but how does it do this? How is charge all of a sudden appearing on the surface?

It is forces of electric field of the battery who induces changes in electric charge distribution in bodies, including the wires. Non-zero surface charge distribution is induced: binding forces of the wire prevent the otherwise mobile charge to jump out of the wire, so it stays on its surface. The equilibrium state is such that this surface charge's electric field cancels out battery's electric field inside the wire.

why doesn't a bulb glow brighter closer to the battery, since the electric field is stronger?

Bulb glow intensity depends on its temperature, not on electric field. Temperature depends on current running through the bulb. When changing position of this bulb with respect to the battery (keeping the wire length the same), this current does not change, because it depends only on voltage on the bulb, and this voltage depends only on electromotive force of the source, and total resistance of the circuit (including the bulb, wires and internal resistance of the emf source).
A: When you connect a battery to a  standard battery + wire + resistor circuit, the battery puts a voltage across that resistor. That is what we say when we do the Ohms law thing.
Well, what is that voltage? The electric potential, as it should be called, is the electric field lines integrated along space. So "1 volt" across the resistor means that the electric field along the space on each side of the resistor is integrating to 1 V (the unit for the E field can be V/m, volt per meter).
The question then is, where did the E field come from? It came from charges. Charges must be building up along the surface of the wire so that the integral of the E field is the required voltage.
You then seem to be asking why the charges leave the battery at all. This is because the battery chemistry sets up an E field between the two terminals, and when the wire is connected to the battery a current must flow in the wire because the field is in the wire.
What this all really means is that when you connect the battery, for a really short time, current is flowing into one side of the circuit, and leaving the other side of the circuit without going through the resistor. Instead, it is depositing on the wire surface on one side, and being taken from the wire surface on the other side. This must happen to create the voltage that eventually drives current through the resistor.
As an aside, this process can be modeled in a circuit without doing a full electromagnetic analysis by simply connecting an extra capacitance across the resistor. That capacitor, sometimes called "parasitic capacitance," is not a real capacitor you connected to the circuit, but represents the capacitance of the actual circuit geometry.
