I feel like the Flat Earth movement is, at it's core, a hack on the human mind.
People want to feel important, and they want to feel in control. When the world is so large and uncaring, so horrible and meaningless, and people are so small and unimportant, a good conspiracy theory lets me be someone important. There are evil people out there who did this all to us, and now I get to know the truth. I get to be in better control of my life. I can keep reading and researching and with each step I feel more in control. I've got hope now because I know what's really going on!
Anyone who challenges this worldview is not attacking my logical argument, they're attacking my emotional investment in this hope, this piece of control of life. When you challenge me, you aren't debating what is and isn't, you're trying to take away this one little bit of control I have. Why would you do that to me? Perhaps you've been brainwashed by the evil people, or maybe you're part of the evil people who did this to us all.