Naively, one is inclined to ask why anyone would actually believe in the non-existence of God. It seems counterintuitive to actively believe that an entity does not exist. This level of commitment might seem to place a burden of proof on strong atheists, leaving weak atheists and agnostics free from such duty*. However, if we distinguish between our model of the universe (i.e. the mechanics of reality as we understand them) from reality itself, the strong atheist position is quite defensible.
Occam's Razor tells us that we should not overcomplicate our models; that we should not posit the existence of entities that do not help explain our observations. It is reasonable to argue that God does not exist in such a model, because God does not help us explain the observable universe. (I use the phrase "it is reasonable to argue", without presenting the argument itself, because here I'm only trying to give an overview.)
This position is roughly equivalent to weak atheism. By saying that you do not believe in God, you are stating (essentially) that your model of the universe currently has no need for such an entity. However, you make no commitment regarding your model of the universe in the future. By contrast, strong atheism might be thought of as the claim that God cannot exist in any useful model of the universe. One can argue this by noting that, excluding gods whose existence is self-contradictory or would contradict empirical evidence, none of the remaining definitions of God are actually falsifiable.
By taking strong atheism to be a statement about our model of the universe, rather than the universe itself, it becomes much easier to support. It transforms the duty of strong atheists from an impossible task - exhaustively searching the universe for a God, to demonstrate his absence - to a much more reasonable one - searching through the various models of God to demonstrate his inapplicability. If God cannot exist in any useful model of the universe, then by definition nobody will ever be able to produce empirical evidence of his actual existence. This is close enough to the statement that "God does not exist" that it makes no odds.
This is all semantics, of course, which is why the debate over strong vs. weak atheism and agnosticism is fairly unimportant. However, if we're going to have such a debate, it's important to realise that the meaning of "existence" (in particular, what it means for something to "not exist") has not been precisely nailed down.
* An aside: atheists do often complain that being asked to "prove a negative" - i.e. the non-existence of God - is inherently impossible. In general this is not the case. I can quite easily prove that there isn't a battleship parked on my front lawn at any given point in time by going outside and taking a photo. Strictly speaking, this isn't "proof" in the sense of being absolutely irrefutable (because, for instance, the photo may be a fake), but it is reasonably good evidence - and that's the best you can hope for when dealing with the real world. The problems associated with "proving" the non-existence of God are far more subtle than simply "proving a negative".