The Social Gospel movement influenced the Civil Rights movement because it applied Christian values and ethics to social issues. Race and the color barrier between white people and black people at the time was a huge issue that this movement focused on, among other issues of course.
As you know, MLK was arguably the most influential public figure in the Civil Rights movement. In a way he was also a big part of the Social Gospel movement in the sense that he was deeply religious, and many of his ideas for social justice were rooted in Christianity.
Hope this answer helps!