The main reason why the Civil Rights Movement fell apart at this point was that it had accomplished the “easy” goals and was now moving on to goals that were never going to be achieved through a social movement.
This is not to say that anything came easy for the movement. Of course, it had to fight long and hard for what it won. But the truth is that the goals it had in the late ‘50s and early ‘60s were much easier to accomplish. These were goals (end of segregation and discrimination, the right to vote) that were perfectly in line with American values. They were also goals that could be easily granted by the government. Finally, these were goals that could be granted without most of white America being directly impacted.
After the Voting Rights Act of 1965, these kinds of goals had all been achieved. Now, the movement turned to things like economic prosperity. For example, Martin Luther King, Jr. was in Memphis to help with a strike by garbage workers when he was assassinated. By this time, the movement was asking for things like economic equality. This was not so much in line with American values. The movement seemed to be asking for equality of results, not equality of opportunities. In addition, the government cannot simply grant everyone economic equality by enacting a single law. Finally, if African Americans were to be given preference for jobs or for admission to colleges, white people would be directly impacted.
Thus, in the late ‘60s, the movement turned to pursuing goals that were much less likely to be achieved. As they failed to have success, the movement fell apart.