Here's a piece called "How Conspiracy Theorists Think." My experience has been that they don't think much but to the extent they do, this piece seems to nail it.
I would also add that I think one reason that conspiracy theorists hang onto the unprovable is that it simply works for them. Some people get money and/or attention for speading nutty ideas about how 9/11 occurred, how JFK was killed, how Obama really isn't President, etc. And they don't want to give that up. I have one guy who sends out a form e-mail every month saying that he's on the verge of proving Hillary Clinton had Vince Foster whacked. He's within an inch of putting Hillary in prison but he only needs a few more bucks for legal fees and won't I please donate? He's been on this same verge for something like ten years now but obviously, each time he sends out this e-mail he gets enough donations to make it worth the few minutes it takes him. So why give it up?