There are at least three reasons why there was a great deal of resistance to this decision.
First, there was the fact of tradition. People typically like to continue to do things in ways that are familiar to them. They do not tend to like change. White children and black children had essentially never gone to school together in the South. This created a strong desire to maintain that system.
Second, there was the fact that the Southern states did not want to be told what to do by the federal government. The issue of states’ rights had been one aspect of the Civil War. The South had also deeply resented Reconstruction. Therefore, the prospect of being ordered to desegregate by the federal government was particularly offensive.
Finally, we have to realize that racism was a major factor. White Southerners simply felt that they were superior to blacks. They worried that integrated education would harm their children by forcing them to fraternize with African Americans. They worried about the possibility of sexual relationships arising between African American boys and white girls. In short, they felt they were better than the African Americans and did not want to be forced to mingle with them on an equal basis.