One way it's been argued that ours is the best possible world is to argue that it is the only possible world. That there is only one possible world, i.e. the actual world, is the thesis I have called monomodalism. It means that nothing at all could be, or could ever have been, otherwise. There is only one real "modality," that of necessity. Of course, if monomodalism is true, then freedom of choice is an illusion. At most, freedom of choice could mean the absence of coercion by other people; but it could not mean that the past and the laws of nature ever permit us to choose other than as we do. In fact, if monomodalism were true, not even the laws of nature could have been otherwise. But natural science affords no evidence of that, and there is no other uncontroversial reason to believe it. Monomodalism, when it turns up in philosophy, is generally a logical excrescence of other pet theses, whose credibility is severely compromised just on that account. The best example of such a system of thought is Spinoza's, which gets a great deal of mileage from stipulative definitions of such terms as substance, mode, attribute, and cause. Einstein admired Spinoza, and he hasn't been the only one. But Einstein still believed in the freedom of the will, and most people who deny they do don't really mean it. If they did, who would be left to blame for anything?
Even so, another, more common reason for thinking ours is the best possible world is that it is created by the best possible being, i.e. God. Now of course it is misleading to talk about God in that way, as if he were just the best and most powerful being there is, rummaging through his immeasurably large file of world-possibilities to select the best one for actualizing. It is better to say, with Aquinas, that God is Being and that "beings" are mere derivatives, existing by a kind of participation in Being. Influenced by the Neoplatonists, especially the Christian Pseudo-Dionysius (5th-6th century AD), some theologians even go so far as to insist that God is "beyond being," so that nothing meaningful can be said about what God is in se. That, it seems to me, makes nonsense of a lot of what such theologians do go on to say about God, but that is a separate topic. The instinct behind their apophatic stricture is sound enough: it just won't do to talk about God as if God were a much bigger, better version of ourselves or any other beings in the world. But fans of the BAPW thesis don't have to talk in that way. They can, and often do, argue that a perfectly good creator can do nothing other than create BAPW. For a perfectly good creator, after all, would be less than perfectly good if he didn't create BAPW.
Over at Prosblogion, Mike Almeida has a nice answer to that. I quote only the formal part of the argument, which is a reductio ad absurdum:
1. Necessarily, a perfect being actualizes the best possible world. Assume for ReductioThe key move in that argument is that, if there were a BAPW, then monomodalism would be true, and hence there would be no freedom. Lacking freedom, BAPW would lack "moral value" and hence not be BAPW.
2. W is the best possible world. Assumption
3. A perfect being exists. Assumption
4. W includes a great deal of natural and moral value. From def. of ‘best world’
5. W is the only possible world. From 1,2,3
6. Everything possible is actual in W. From 5
7. W is a fatalistic world. From 6
8. No moral agent is libertarian or compatibilist free in fatalistic worlds. Fact
9. No moral agent is free in W. From 7, 8
10. There is no moral value in W. From 9
11. W is not the best possible world. From 10, 4. Contradiction 11,2
∴ 12. It is not necessary that a perfect being actualizes the best possible world. From 11,2
I think that pretty much hammers the last nail in the coffin of BAPW as a thesis. And that's important theologically inasmuch as, if there could be no BAPW, then God can't be blamed for failing to create it.