there seems to be no rigorous sense of what "requiring an explanation" means in the first place.
"Requiring an explanation" means "low probability". An "explanation" is a datum such that conditioning on it makes the probability high.
You can think of probability as an "inverse surprise score" that you try to keep as high as possible. (And of course, there's no cheating.)
negative surprise log score?
Does the Universe Need God? (essay by Sean Carroll)
In this essay, Sean Carroll:
Dissolves the problem of "creation from nothing":
Uses Bayesian reasoning to judge possible explanations:
Correctly describes parsimony in terms of Kolmogorov complexity:
Discusses "meta-explanatory accounts":
Points out the theory-saving in and the predictive issues of God as a hypothesis:
See also his blog entry for more discussion of the essay.
Edit: added the bullet point about "meta-explanatory accounts."