I have a car with a faulty alternator. My battery went flat. I couldn't start my car and left it for an hour. I tried again after an hour and it started. How did my battery re-charge itself?

It didn't recharge itself, sound like you have a bad starter switch (relay) or possible the starter is no good. when the car heats up that when the problem is showing up and when it cools down it allows the car to start again.

Also, once you get the car started, go somewhere to have the charging system checked while the car is running, As joemorn said, possible bad starter, worn bushings in the starter will cause the starter to bog down when warm.