Like many terrorist organizations, the KKK is an insurgent group that came post a defeat in a major war. It was started by defeated confederate generals who had to switch to Plan B after their loss in the US Civil War [1861–64]. In this, it is similar to the Islamic State that rose post the war in Iraq.
In its first incarnation in the 1860s, in the rural parts of US south it was used as a way to continue the slavery imposed on the black population. The erstwhile slave owners could not digest the idea of emancipation and used various terror methods to continue the slavery.
It survived for a decade, but the actions of the federal government put an end to it in the 1870s.
About 40 years later the KKK rose in the wake of the first world war. The chaos in Europe sent droves of refugees toward the US from the late 19th century. These included Jews and Catholics, who were often seen with some hostility by some of the natives. The immediate catalyst for the second founding was the famous film The Birth of a Nation that glorified the clan movement. Like dog whistle it pulled a new generation of clan members, with one key difference that this became more of an urban movement. They especially grew strong in the prohibition era and used their raw power to intimidate anyone connected to liquor.
Once again, the clan survived strong for a decade and petered out. This time from the action of bold journalists.
The third incarnation began in the 1950s, when the blacks started to demand civil rights. The clan once again came into force as a way to intimidate them and provide some assurance to the supremacists that things stay unequal. This time the clan was probably more tolerated by the government, as they also had their wrath on communists - the bug bear for the US elites. The FBI allegedly went soft on the clan due to their focus on the “commies”.
The KKK like many terrorist organizations around the world is a way to use the power of intimidation towards political goals. In various incarnations it included the goals to continue slavery, prevent immigration, attack alcohol [liquor was especially sought by new immigrants like the Italians] and later to impede civil rights.