Or maybe the better question would be, why did it start in the 50's?
The civil rights movement did not start in the 1950s. Rather, that's what a lot of historians call the "mainstream" civil rights movement. And to be fair, for quite some time historians really just focused on MLK, Rosa Parks and the like, so the idea that the civil rights movement sprang up out of nowhere in 1954 is remarkably durable. There has always been some kind of civil rights movement in the United States, even before there was a US. As long as black people have been oppressed in some way in the Americas, there has been some sort of movement to protest that treatment.
Here are some sources/suggestions for further reading:
Biondi, Martha. To Stand And Fight: The Struggle for Civil Rights in Postwar New York City. Harvard, 2006.
Foner, Eric. Forever Free: The Story of Emancipation and Reconstruction. Vintage, 2006.
Foner, Eric. Reconstruction: America's Unfinished Revolution, 1863-1877. Harper, 1989.
Gilmore, Glenda. Defying Dixie: The Radical Roots of Civil Rights, 1919-1950. WW Norton, 2009.
Gilmore, Glenda. Gender and Jim Crow: Women and the Politics of White Supremacy in North Carolina, 1986-1920. UNC Press, 1996.
Hahn, Steven. A Nation Under Our Feet: Black Political Struggles in the Rural South From Slavery to the Great Migration. Belknap, 2005.
Johnson, Walter. Soul By Soul: Life Inside the Antebellum Slave Market. Harvard, 2001.
McCurry, Stephanie. Confederate Reckoning: Power and Politics in the Civil War South. Harvard, 2012.
Nash, Gary B. Race and Revolution. Rowman and Littlefield, 1990.
Payne, Charles. I've Got The Light of Freedom: The Organizing Tradition and the Mississippi Freedom Struggle. University of California Press, 2007.
Oakes, James. Freedom National: The Destruction of Slavery in the United States, 1861-1865. WW Norton, 2012.
Satter, Beryl. Family Properties: Race, Real Estate, and the Exploitation of Black Urban America. Metropolitan, 2010.
Wilkerson, Isabel. The Warmth of Other Suns: The Epic Story of America's Great Migration. Vintage, 2010.