This is a great question, and I learned a few things from the other replies here—especially the one by Desmond Ng. I don’t have any special knowledge about flat-earthers, but I spent many years reading letters to the editor at two major science magazines (Scientific American and Discover). There were two common threads to most of the conspiracy-theory letters we received:

* A sense of smug superiority. It often is expressed in some version of “I’ve got my eyes wide open and I see what’s really going on.” Belief in the conspiracy makes the believer feel special, awake, smarter than everyone else. That feeling can exist only if the conspiracy exists, so here is the important point: There does not need to be any rational reason for the conspiracy to exist. The deep desire is for the conspiracy to be real, and anyone with a deep desire can come up with endless reasons why that desire might be real. It’s somewhat like convincing yourself that someone you desire is secretly madly in love with you, despite all signs to the contrary. There are people who do that, too.

* An intense paranoia that “They” are laughing at the ones being fooled. This is the flip side of the previous feeling. In the mind of the conspiracy believer, the world is upside down and the people running the conspiracy are the ones enjoying a sense of smug superiority. There is a huge projection effect going on here, as you can see. The believer feels such a rush from being in on the secret that he/she is sure that the keepers of the conspiracy must feel the same way. Sure, there are always all kinds of reasons given for what makes the conspiracy valuable, but at the deepest level the true believer assumes that the conspirators keep their secret simply because they enjoy keeping the secret. The brilliance of this mindset is that no kind of rational argument can ever budge it.