The Civil Rights Movement succeeded in at least two main ways. First, it succeeded in that it won more legal rights for African Americans. Second, it succeeded because it helped to create a less racist society, one where whites and blacks get along better (though by no means perfectly).
The clearest success of the Civil Rights Movement is the fact that African Americans now have all of the same rights that white Americans have. Before the Civil Rights Movement, segregation and discrimination were legal. It was also legal for states to engage in a variety of activities that prevented African Americans from exercising their right to vote. The Civil Rights Movement was aimed in large part at doing away with these injustices. With the Civil Rights Act of 1964 and the Voting Rights Act of 1965, this aim was accomplished. Segregation and discrimination are now illegal and states cannot use things like literacy tests to prevent African Americans from voting. Before the movement, African Americans had very few rights. Today, they have all the rights that other Americans do. This is the main reason why we can say that the Civil Rights Movement was a success.
The other way in which the Civil Rights Movement succeeded is less obvious and more debatable. However, I would argue that the movement succeeded by making American society less racist and more tolerant. Before the Civil Rights Movement, it was perfectly acceptable to be openly racist in much of American society. In most parts of the United States, very few white and black Americans associated with one another as equals. It would have been unthinkable, for the most part, to have black teachers teaching white students, black police arresting white people, and black government officials helping to govern white constituents. Today, none of these things are remarkable. Today, there are very few places in the United States where it is acceptable to voice openly racist opinions. This is not to say that there is no racism, but there is much less than there once was. This is, in my view, a way in which the Civil Rights Movement succeeded.
I would argue, then, that the Civil Rights Movement succeeded because it won legal rights for African Americans and because it helped to create a society that is less racist and more just than the society that existed before the movement.
For more information on the history of the the Civil Rights Movement, check out this video: