Let's break this down a little bit. The context here is Stephens talking about how wrong the founding fathers had been. He framed them as either anti-slavery or at least uncomfortable with the institution. In his view, they had imbibed harmful enlightenment ideas about the equality of all human beings. And though their Constitution had protected racial slavery well enough for three generations, it was now time for the South to break away and establish itself firmly on the foundation of racial slavery, in perpetuity. Rather than seeing slavery as a necessary evil, it was now a positive good. The southern system was not only practical as a means of creating prosperity and White social solidarity, it was a divine order aligned with natural and moral law.
Our new government is founded upon exactly the opposite idea [of the equality of all human beings]; its foundations are laid, its corner- stone rests, upon the great truth that the negro is not equal to the white man; that slavery subordination to the superior race is his natural and normal condition. This, our new government, is the first, in the history of the world, based upon this great physical, philosophical, and moral truth...it is, indeed, in conformity with the ordinance of the Creator. It is not for us to inquire into the wisdom of His ordinances, or to question them. For His own purposes, He has made one race to differ from another, as He has made “one star to differ from another star in glory.” The great objects of humanity are best attained when there is conformity to His laws and decrees, in the formation of governments as well as in all things else. Our confederacy is founded upon principles in strict conformity with these laws. This stone which was rejected by the first builders “is become the chief of the corner” the real “corner-stone” in our new edifice.
Alexander Stephens, Vice President, CSA
Christians will immediately recognize that Stephens' speech is not only explicitly racist, but blasphemous. Using language the New Testament uses repeatedly to describe Jesus Christ, Stephens calls slavery "the stone" which was rejected by the first "builders" (the founding fathers). White southerners have now made slavery the "corner-stone," building their whole society upon it. In the New Testament, Christ is the cornerstone of the household of God, the cornerstone of the church (Ephesians 2 and elsewhere). For Stephens, slavery is to the Confederacy as Jesus Christ is to the Christian church. You comfortable with that?
Where, then, did all this states' rights stuff come from? It has a variety of roots, but one of the funniest is Stephens himself. By 1868, three years after losing the war, he was already singing a different tune. He wrote that the war had merely been a battle between contrasting views of government; one national, one federal. It wasn't about slavery after all! Unfortunately for Stephens, he was unable to expunge his 1861 speech from the historical record.
The reason the "cornerstone" speech is important is not because it is a "gotcha" document to use as a cudgel in a historical debate. After all, you can find people saying all sorts of things in the historical record, much of it contradictory. Rather, the speech is useful precisely because it exemplifies, in a particularly clear and entertaining fashion, what you'll find to actually be true of the antebellum South if you look into it a little bit. The South really was a slave society on an intense scale, one of the few in global history. It was a slave society in that not only did it have enslaved people (as many societies all over the world have) but one's identity, one's social place, was based primarily on one's relationship to the institution. There were three primary kinds of people: enslavers, enslaved, and those hoping to become enslavers. (How free Blacks in the South fit into this picture is an interesting discussion.)
Last I looked, some polling indicated that nearly half of Americans think the Civil War was about states' rights. If they're really attached to the term, they can keep it I suppose, as long as they correctly answer the follow up question: a state's right to do what?
All of this often feels rather trivial or even silly, but in fact these false beliefs are a mark of the moral degradation of our public culture. Good and decent people harbor these assumptions, which indicates a systemic problem. It is not a question of personal evil. This kind of systemically enforced ignorance is similar, I imagine, to the genocide and atrocity denial that is mainstream in places like Turkey and Japan. I firmly believe public memory matters. If we can't be honest about the White supremacist state our founders established, and the myriad ways it survived civil war and Reconstruction, we certainly are not equipped to repair it.