According to the PBS article in this link, the Great Society was a set of programs and policies that President Lyndon B. Johnson proposed in an attempt to create a society
in which America ended poverty, promoted equality, improved education, rejuvenated cities, and protected the environment.
We can see that civil rights legislation, like the Civil Rights Act of 1964 and the Voting Rights Act of 1965, would have helped to promote some aspects of this agenda.
The Civil Rights Act of 1964 outlawed racial discrimination in public accommodations. That means, for example, that it was illegal to discriminate against African Americans in hiring. This could conceivably help reduce poverty because it would make it easier for blacks to get jobs. It would clearly promote equality because it removed laws that discriminated against African Americans.
The Voting Rights Act of 1965 ensured that states could not limit the right of African Americans to vote. This would have helped promote the goals of the Great Society because it would give African Americans more political clout. If blacks were able to vote, politicians would have to take actions that would appeal to them. This would make politicians more likely to enact laws aimed at reducing poverty, improving education, and other goals of the Great Society.
In these ways, civil rights legislation can be seen as a cornerstone of the Great Society.