The best way I ever had it explained to me was that if you imagine a swimming pool full to the top with ping pong balls, and you push one extra ball in at the shallow end, another ball will pop out at the deep end (well, it's a simplified logic-problem view). This is how the electron flow works: one electron will leave the negative terminal of the battery (due to the voltage, or potential difference along the length of the circuit) and enter the wiring. This wire is made up of molecules of copper; the electron will strike a molecule and displace another, which will then strike another molecule, following the flow around the circuit. The original electron itself may take a very long time to propagate around the circuit (or that specific ping pong ball you put into the shallow end may take a long time to leave the deep end). But the total effect of one-in-one-out is preserved.
Put another way, the wire is already filled to capacity with electrons (or the pool with balls) before the switch is closed. Adding balls/electrons at one end (almost) instantly pushes one out the other, but the actual electrons themselves can move very slowly from one end to the other.
Going back to the hydraulic analogy, your house is likely connected to a municipal supply (or dam, bore, whatever) through tens, or hundreds of kilometers of pipeline. Inside this pipe is water. With the tap in your house turned off, the water is under pressure (voltage) within the pipe due to the pumps at the supply/elevation of the dam. When you turn the tap on, water instantly begins to flow. However, this is the water (electrons) which were already in the pipe. If the dam/pump added a red dye to the water, it may take hours for this dye to appear out of your tap, as all of the 'clean' water (electrons) just first be removed from the pipe.
That is an extremely generalised explanation and doesn't cover things like valence bands (electrical conductivity of the molecules) or semiconductor effects (molecules becoming positively charged as they lack an electron) but it serves the purpose as far as explanations go. If you'd prefer a more detailed explanation, I'd recommend obtaining a copy of Neamen, DA 2010, Microelectronics: circuit analysis and design, 4th edn, McGraw Hill, London. Which was the undergraduate text I used, and it goes into all the detail you're likely to need about the 'how'.