During the Cold War, the US and the Soviet Union were competing for influence around the world. After they pretty much split Europe up, they had to compete for influence in Africa and Asia. Both of those areas are largely made up of non-white people. It was for this reason that America's Cold War role helped create a climate for changing race relations.
It was embarassing for the US to have segregationist laws at home (even in Washington, D.C.) while it was trying to get a lot of non-white people to like the US. It made those people think that allying themselves with the US might not be the smartest thing ever. The Soviet Union tried to portray itself as a non-racist alternative to the US. Because of things like this, many in the US who were mainly concerned with containing communism also came to want race relations in the US to improve.
So, the American role as defender of democracy created a climate for improved race relations because the US needed to look better to the non-whites of the world.