D.W. Griffith's views on Abraham Lincoln and the KKK are thoroughly identified with the "Lost Cause" interpretation of the Civil War and Reconstruction- namely, that the South was a virtuous society that fought for noble causes, downplaying the role of slavery, and was defeated because of overwhelming disparities in resources and manpower, and because some of its generals bungled the best laid plans of persons such as R.E. Lee. The "Lost Cause" movement also diagnosed the Reconstruction era following the war along similar lines. The idea that perfidious radical republicans had enacted harsh vengeance against southerners and gave their rightful place in government to former slaves and northern transplants is a major component here. Many in D.W. Griffith's time believed that Jim Crow was the proper nature of their society and that had Lincoln lived, or had Johnson not been undermined by the radical wing in Congress, the era of "Redemption" and "good feeling" could have been arrived at sooner. Ergo, in Griffith's film, Lincoln is portrayed as a moderating and sympathetic figure, made to stand in contrast to the spiteful Thad Stevens.

After the war, the North fully threw itself into the task of reintegrating the southern states, but the question of how best to do this was vociferously debated. Should the states be quickly reintegrated with as little fuss as possible, or did the secessionist south require some sort of wholesale re-organizing? The Republican Party was the primary arbiter of these debates (as it essentially controlled the Federal government), and it had two distinct wings. The moderates, initially led by Lincoln and then by Andrew Johnson after Lincoln's death, held that once 10% of the 1860 voting populous affirmed loyalty to the Union and accepted the end of slavery, that state should be reintegrated in the USA. The Radical Republicans (as they were known) favored more thorough measures for re-integration, such as harsher punishments for leading secessionists, strong legal protection for former slaves via protection by Union troops and the Freedmen's Bureau, and possibly treating the defeated Confederate states as conquered provinces rather than states. Furthermore, they wanted to (and did) give the ballot to African-Americans, as well as empowered them with the ability to run for and win major offices in these southern states. This was anathema to the white south. The two Republican factions clashed while Lincoln was president over the conduct of the war and the future of reconstruction, but while the war was still being fought these differences weren't as explosive, or were at least shunted aside while the task was still at hand. By the summer of 1865, however, the two sides devolved into completely bitter disputes as the war ended and Johnson proved less politically adept than Lincoln and far more conciliatory toward the south than he had first appeared to be. Johnson's presidential construction gave way to congressional reconstruction and then to Johnson's impeachment proceedings.

The former CSA states, as you can guess, favored the moderate approach as championed by Johnson. They strongly resented the radical wing of the Republican party imposing its will through Congressional Reconstruction on them. They did everything they could to subvert congressional efforts at reconstruction and formed vigilante groups, including the KKK, to terrorize local blacks who exercised their new rights and those whites who helped them to do so. It often took northern troops stationed in the south to countermand these groups. After the incredibly close and contentious election of 1876, and over a decade of attempted reconstruction of the south, the north essentially lost its political will to continue the massive undertaking it had begun and allowed the white south to fully implement Jim Crow and reassert itself as it wanted to.

This was an exceptionally bitter, tragic (especially regarding the brief respite and cruel undercutting experienced by African-Americans), and polarizing period in American history. Many in the South grew to accept moderate Republican views espoused by those such as Lincoln and Johnson as wise (or at least much more favorable to them than they were going to get elsewhere). Conversely, they loathed the radical republican agenda completely.