Madison defined our democratic government as a republic rather than as a pure democracy. The difference, in his words from The Federalist Papers, is that a republic is "a government in which the scheme of representation takes place." The Civil Rights movement advanced the struggle to achieve American democracy by fighting for African-Americans to enjoy the same basic liberties as other people and by gaining access to the franchise (the right to vote) for African-Americans.
Before the modern phase of the Civil Rights movement, African-Americans' rights were restricted. In the south, very few African-Americans were allowed to vote, as they were subject to poll taxes, literacy tests, and grandfather clauses (which stated that people couldn't vote if their grandfathers couldn't vote). In some southern states, the African-American voting rate was less than 10%. The Civil Rights movement fought for African-American voting rights. For example, in 1964, the Mississippi Freedom Summer, organized by CORE, Martin Luther King's SCLC, SNCC, and the NAACP, registered African-Americans to vote. Civil rights workers were killed, and others risked their lives. Eventually, the Voting Rights Act of 1965 put the power of the federal government behind the enforcement of African-American voting rights. By being able to vote, African-Americans were able to exercise their right to participate in our republican form of government by which we elect people to represent us at the local, state, and federal levels.
The Civil Rights movement also resulted in the Civil Rights Act of 1964, which made racial and forms of other discrimination in public facilities, schools, and the workplace illegal. A democracy allows rights to all its people, including people in the minority, so by granting these rights to African-Americans, the U.S. became more democratic.