First, people put their identities into their beliefs. Everything is a lifestyle choice anymore. What non-mainstream belief defines me as a person? When you make believing in the flat earth part of your identity, you get a fight or flight response when someone attacks the idea of the flat earth, because in a way, they are attacking you.
Second, it's easy to confirm your misinformation with more misinformation! I don't think people hear something once, like "the earth is flat" and just believe it. They hear it, look it up, see multiple sources saying the same things (weren't we taught to confirm stuff like this?), sprinkle in the smallest amount of plausibility - maybe a source that otherwise prints stuff you know to be true (sports scores, natural disasters, whatever) also prints something about the flat earth, and there you have it. You've done your homework. At this point admitting you're wrong means you must admit you are ignorant and were so easily duped, or that your entire education was a waste of time. It takes a very big person to do that.