We are not told for certain why the Ku Klux Klan wanted to kill these three men, but Holmes has a theory. He notes that the KKK started to fall apart right after the time that Colonel Elias Openshaw left the United States. This was in 1869. Holmes says that Elis took the KKK's papers with him when he left. The papers could have shown that various important Southerners were part of the KKK.
So the KKK seems to want to get revenge on Elias for taking their papers and harming their movement. Since they are so ruthless, they want to punish others in his family as well. That is why they kill his brother and nephew.