In the Civil Rights Movement, important things were at stake (or were felt to be at stake) for both African Americans and whites, particularly from the South. For the African Americans, what was at stake was legal equality. For whites in the South, what was at stake was their traditional social order.
African Americans had never been legally equally to whites in America at the time of the Civil Rights Movement. The point of the movement was to rectify that situation. African Americans and their supporters wanted to make sure that blacks would be equally to whites in the eyes of the law.
Because African Americans had never been legally equal to whites, Southern society was based on inequality. Generations of Southern whites had been taught that this was normal and just. When people have lived with a given system for a long time, they become attached to it. They worry about what would happen if things change. Since they like the way things are, they do not want it to be changed. Change threatens people who are content with the status quo. This is why Southern whites fought so hard against civil rights. They worried that it would destroy the system that they had lived in for decades and which they felt was a good system. It is not strange that people would fight hard to prevent this from happening.