The FAQ has a couple comments you might find useful. In particular /u/Georgy_K_Zhukov's comment about the Washington Irving and the myth of Columbus proving the world was round. This myth of the flat earth became a favoured polemical devise of the anti-religious side of the culture wars of the mid to late nineteenth century and naturally proliferated in this context.

As to why it has stuck around. I think there are a few reasons that we can point to. In the first place, the very conception of modernity depends on its opposite, the middle ages. Though we do not live in quite the age of religious controversy as in the 19th century, between a newly developing academic atheism and the reemergence of academic Catholic apologetics, we still very much ground our self-conception in terms of historical antecedents and contraries. In particular, the renaissance/enlightenment notion of the classical world as our true cultural predecessor, interrupted by the [insert appropriate pejoratives] medieval world. On top of this, I think many people have an essentially positivist view of history à la Auguste Comte. Thus they view the history of the world in terms of the inexorable march of science, somewhat in line with Comte's three three stages of civilisation: childhood, youth and adulthood. The childhood of civilisation is characterised by the theological mindset, according to which the basic nature of things is explained by the worldly activity of polytheistic and monotheistic entities who populate the cosmos. The youth stage is metaphysical, so these entities become abstractions, and the questions though still theological become about principles rather than things. The final adult stage is the, so called, positive stage of civilisation. At this point they have grasped the vain nature of searching for abstract metaphysical principles and turn to the empirical sciences. An implicit premise of this narrative is the scientific backwardness of previous civilisations, so when people hear that ancient or medieval people thought that the earth was flat, this is the sort of proposition they expect, thus it is readily accepted.

There is a second important reason that has to do with the nature of the scholarship on medieval geography and cosmology. Medieval geography (under which this sort of thing is normally discussed) is deeply understudied. So much so that, particularly in English, J. K. Wright, The geographical lore of the time of the crusades (1925) is still a standard work on medieval geography and both C. R. Beazley, The Dawn of Modern Geography (1890) (eg. 1.274f) and G. H. Kimble, Geography in the Middle Ages (1938) are not far off. Beazley in particular was tremendously influential on the development of the modern study of historical geography. All three of these works, though showing some scholarly nuance on this point, present the view that some or the majority of Church Fathers and Medievals believed that the earth was flat and tend to treat belief in the sphericality of the world as an exception rather than a rule. Likewise, Wright and Kimble both suggest (incorrectly) that belief in a round earth was at some point heretical. (For example, you can read the section in Wright (p. 53f) here and likewise in Kimble, 35f (you can borrow this on archive.org).) Due to a combination of laziness and perhaps prejudice, these views have been repeated by scholars who don't specialise on the middle ages, as in Daniel Boorstin's 1983 The Discoverers: A History of Man's Search to Know His World and Himself.

Edit:

Just for some further reading, I've essentially followed Patrick Gautier Dalché, La terre: connaissance, représentations, mesure au Moyen Âge (2013), 166-70. However, for a book length treatment of the issue see Jeffrey Burton Russell, Inventing the Flat Earth: Columbus and Modern Historians (1991).