Well if you apply the general 'how it works' from any electochemical cell, I'd assume that's how it works.
So there's a reaction between the lead anode and the sulphuric acid, which is the oxidation reaction. Here's where the electrons come from. They travel through the external part of the circuit (wires and stuff) [from the prelim topic metals, you should know how electricity is conducted in the wire] and eventually reach the cathode of lead dioxide. They are accepted here for the reduction part of the reaction.
Here's where I'll have to assume that it follows the same principal as other cells. The sulphate anions are travelling to the anode, and the hydrogen cations to the cathode. This completes the circuit, allowing current to flow.
I hope that answers your question.
Feel free to correct any mistakes I've made.