From a quick Google, almost everyone who discusses the history of racial segregation on the Internet only mentions the arguments against it. It's certainly a good sign that segregation is so widely opposed now. But Americans in the 1950s weren't all idiots. Surely, during years of protesting and debating, someone on the segregationist side argued their case in more detail than "black people are bad". What were the major arguments of the segregationists, and how were most people ultimately convinced that they were wrong, when they had (apparently) been accepted for decades before?
George Wallace the Governor of Alabama, famous for saying "Segregation now, Segregation forever" in his inauguration speech in 1963, viewed segregation as "best for both races" because they "prefer their own churches and schools". He saw civil rights leaders like MLK as outside agitators trying to disrupt the rule of law and the peace that segregation created. You can read more him and his speech here.
But these ideas didn't come out of nowhere. If you want to learn about the roots of these racist ideas I think you have to go back to the 19th century. Many scientists and early anthropologists put forth theories that blacks were a different species to excuse slavery. Once slavery was made illegal segregation was promoted and popularized as an idea to prevent miscegenation or race mixing. I would highly recommend W.E.B Du Bois excellent book Black Reconstruction in America if you want to read up on this.