Swinburne's explanatory argument is basically that while Naturalism cannot explain the origin of minds (e.g. he asks, "how far could the science of the future explain the evolution of souls," in his 1997, p. 174), Monotheism can explain the world's evils (e.g. via my theodicy), whence the latter is the best explanation... but of course, that is so only if Monotheism can explain the origin of souls (otherwise it would be rather like postulating a pork chop to explain crop circles).
......In particular, Swinburne postulates a perfect person as a simple hypothesis (with a correspondingly high prior probability) to explain the world, but that is only explanatory if creating worlds is conceivably something that a person could do. Now, we know that people can rearrange things, but so can evolution. We know that people dream (can even deliberately daydream), so maybe people can create lower sorts of being (if dreams have a sort of being-in-themselves; if we do indeed create them, rather than just experience them), whence a transcendent person might create a physical world; but if God makes people like us (as souls, with free will) as a lower sort of being (not just an imperfect or finite sort), then we seem to lose our sense of God as a perfect person, and the high prior probability without which the God hypothesis isn't even contending...
We seem to, but the criterion of simplicity itself stands in need of justification (or explanation). Maybe what we really want are, to begin with, a few hypotheses that promise to be worth looking into and which seem fairly exhaustive (the alternatives being clearly too odd); hypotheses that we can hope to work with easily enough (simplicity), which promise enough of a pay-off (if they win) and which are naturally unweighted to begin with (prior to the evidence). I'm not suggesting that a big reward for belief can make us believe (Pascal's Wager is off) but after all, the reason why we value truth that highly is, under Naturalism, that genes for such valuations were rewarded, with reproductive success (while under Monotheism truth relates us to God; and incidentally were we made in God's image, simpler concepts would be more likely to resemble those behind creation).
......So Monotheism remains a contender... And while creation remains mysterious, so do mind and matter given Naturalism; and if it's conceivable that matter could spontaneously appear in a Big Bang, and that minds like ours could (somehow) arise from such material, then surely it's similarly conceivable that a sufficiently unlimited person could cause such things deliberately (and incidentally the substance dualism that is our common sense experience is more explicable given Monotheism).