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As a Catholic, I try to approach Scripture as it has been interpreted by the Church rather than try to interpret Scripture for myself.  I know that most of Catholics throughout history (at least the ones we know about) believed that the earth is a sphere.  I do not want to take a position different from that of great Catholic minds like St. Bede, St. Thomas Aquinas, and St. Robert Bellarmine.  These brilliant and holy men accepted St. Augustine's teaching that Scripture is silent about the shape of the earth. They believed it was a matter of science and that science shows that the earth is a sphere.  Their arguments for a spherical earth make sense to me, but I would be reluctant to disagree with such men even if I could not follow the arguments.

The idea that the earth is flat had a few supporters in the early centuries of the Church, but it disappeared for more than a thousand years.  Belief in flat earth was revived in recent times by Protestants interpreting Scripture for themselves without reference to Catholic tradition.  There is no reason for Catholics to accept this belief.

It's just my opinion, but I think that they are afraid. Afraid that trads will look stupid. It's normal for humans to care about what others think, but the more I look at the lives of the saints, the more I realize that they cared more about what God thinks, rather than being so concerned with human respect. And lets face it, trads are more than "a little out there" already. Why add to the nuttiness by believing in a flat earth?

It is legitimate to be concerned about how trads proclaiming that the earth is flat makes all trads look bad.  St. Augustine wrote this: 

``Usually, even a non-Christian knows something about the earth, the heavens, and the other elements of this world, about the motion and orbit of the stars and even their size and relative positions, about the predictable eclipses of the sun and moon, the cycles of the years and the seasons, about the kinds of animals, shrubs, stones, and so forth, and this knowledge he holds to as being certain from reason and experience. Now, it is a disgraceful and dangerous thing for an infidel to hear a Christian, presumably giving the meaning of Holy Scripture, talking nonsense on these topics; and we should take all means to prevent such an embarrassing situation, in which people show up vast ignorance in a Christian and laugh it to scorn. The shame is not so much that an ignorant individual is derided, but that people outside the household of faith think our sacred writers held such opinions, and, to the great loss of those for whose salvation we toil, the writers of our Scripture are criticized and rejected as unlearned men. If they find a Christian mistaken in a field which they themselves know well and hear him maintaining his foolish opinions about our books, how are they going to believe those books in matters concerning the resurrection of the dead, the hope of eternal life, and the kingdom of heaven, when they think their pages are full of falsehoods and on facts which they themselves have learnt from experience and the light of reason? Reckless and incompetent expounders of Holy Scripture bring untold trouble and sorrow on their wiser brethren when they are caught in one of their mischievous false opinions and are taken to task by those who are not bound by the authority of our sacred books. For then, to defend their utterly foolish and obviously untrue statements, they will try to call upon Holy Scripture for proof and even recite from memory many passages which they think support their position, although they understand neither what they say nor the things about which they make assertion. [1 Timothy 1.7]'' 

Concern for human respect is a problem when we place it above Catholic teaching.  When it comes to flat earth, people merely have a personal opinion that they are attached to.  They would do better to refrain from promoting it since St. Augustine calls such behaviour "a disgraceful and dangerous thing."

I personally do not accept the idea of flat earth because it is not compatible with Catholic history and tradition.  However, if there are people who are afraid that it makes us look stupid, they are following the teaching of St. Augustine.