After the Civil War, during 'Reconstruction', many blacks were elected in the South, something which took many decades to happen again. You often here of someone as being the first black elected to X Southern State since Reconstruction.
So, how did those people get elected? Were the Republicans stealing the election from the whites, and getting black Republicans elected? The movie implies that the KKK was formed in response to this very thing. This is why the KKK is so tied to the Democrat Party--it was formed to get white Democrats elected.
It's complicated, no one is innocent, but I think this movie, based on the date it was made, could very well have more truth in it than the offical story.