What role did racial issues and state’s rights play in the evolution of federalism in the post-Civil War era of the late 19th century?

You may or may not recall, but the only the first twelve amendments to the Constitution existed at the end of the Civil War.  These amendment, the Bill of Rights, plus two more, along with the original Constitution, were rights that protected the people from the federal government.  But there was no protection from whatever state governments chose to do.

After the Civil War, since states were not constrained by the Constitution or its amendments, they were able to deprive people of rights with impunity. And the Southern states proceeded to do so, out of anger over the war and prejudice against the now freed African-Americans. They made it impossible or nearly impossible for freed slaves to vote, own property, or be treated like equal citizens in any way whatsoever.  The statutes passed in the Southern states are often referred to as "Jim Crow" laws.

These laws led the federal government to realize that winning the Civil War was a hollow victory if the Southern states could continue to deprive free people of all or most of the attributes of freedom.  Thus, the next amendments established federal control over various freedoms, protecting particularly black American not only from the federal government, but also from the state governments. 

I have provided a link to the Constitution and its amendments.  Look carefully at the dates of the amendments after the Bill of Rights, and you will see how and when federalism evolved.