What do people actually mean by any of this? Do they actually conduct their lives on the working assumption that the next stranger they meet may be an angel, a ghost, Satan, or a UFO crewman? (Ans: Obviously not.) How many could give a coherent account of the theory they reject? (Ans: Vanishingly few.) What does “believe” actually mean in this context? (Ans: Nothing very functional.)Douthat's answer is, I think, very much on base. He concludes:
But if you want to understand what, if anything, a person means when he says he believes in demons or angels or ghosts, the simplest baseline answer is this: He means that if confronted with an encounter or an experience that seems demonic or ghostly or angelic and asked to rationalize it, he will be inclined to give credence... to the possibility that the encounter is, in fact, what it appears to be.I think this is one of the basic areas in which materialists and non-materialists talk past each other a great deal. There's a sort of creation myth of mythology, religion and folklore that moves about in secularist/materialist circles (perhaps given that these are materialist circles we really should use Dawkins' terminology and call it a meme) that all forms of "superstition" are in fact attempts explain how the world works. Thus, belief in God is really just an attempt to explain "how things got here" in the absence of scientific knowledge on the topic. Belief in angels or demons or ghosts or any number of folkloric characters is simply a way of explaining phenomena which people didn't understand the physical causes of. Or perhaps a way of reading causation into something which is actually random -- an over-active sense of pattern recognition.
However, I think part of the real difference here is that people have strongly different ideas about how unlikely it is to experience something that is non-material. For some people, the idea that the sun danced at Fatima or that late stage cancer could completely vanish as a result of miraculous means is so unthinkable that even if one receives multiple testaments to such a thing having happened, one concludes that there must be some completely unknown (or highly unlikely) physical explanation which is what "really happened."
But the difference is not so much between people who look for "real explanations" for things and people who don't, but rather between people who discount all non-material explanations and those who don't.
Needless to say, there is still plenty of room for disagreement. I've often had fellow Christians relate to me small "miracles" which I would tend to assign as simply being coincidences. But again, that's a difference primarily attributable to my placing a different likelihood on supernatural intervention in certain issues than some other people do.
One of the primary charges often leveled against religious believers is that they are irrational -- but I think the key difference to understand here is that believers in various forms of non-material creature or phenomena do not fail to understand cause and effect, or fail to look for rational explanations, it's simply that their ideas of what is a possible cause for a given effect are difference from those of the committed materialist. Whether the materialist's commitment to all phenomena having material explanations is justified by anything other than choice is less clear to me.