It makes Flatties feel special. To them, they've "cracked into" some terrible conspiracy that they can fight valiantly against. A conspiracy that's so large that literally anyone can be in on it, which is appealing to the Flattie because it makes the fictional conspirators a big bad guy for them to fight. This is why they're impossible to convince to even consider that their notion that the Earth is flat is wrong because to them they're already convinced of "The Truth" and that anyone who says otherwise is a henchman sent by the big bad guy to challenge the hero (the Flattie). The best one can hope for in attempting to debunk the Flat Earth Theory is to show to outsiders that the theory has no foundation and that it shouldn't be considered thus saving someone from becoming a Flattie.
TL:DR - Flatties accept that the Earth is flat because it makes them feel like Neo in The Matrix, when in reality they're more like that old lady in The Happening.