How does South Carolina's secession in 1860 show that slavery was at the heart of the American Civil War?

On April 26, 1854, South Carolina, as part of the newly formed Confederate States of America, issued the "Declaration of the Immediate Causes Which Induce and Justify the Secession of South Carolina from the Federal Union."
Initially, the declaration cites the violation of its states' rights, as guaranteed by the Tenth Amendment, as cause for secession:

By this Constitution, certain duties were imposed upon the several States, and the exercise of certain of their powers was restrained, which necessarily implied their continued existence as sovereign States. But to remove all doubt, an amendment was added, which declared that the powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States, respectively, or to the people. On the 23rd May, 1788, South Carolina, by a Convention of her People, passed an Ordinance assenting to this Constitution, and afterwards altered her own Constitution, to conform herself to the obligations she had undertaken.

Some people prefer to cite states' rights as the primary, or sole, cause of secession, choosing to look at slavery as a side issue in relation to the more central issue of concerns over federal power. However, this view overlooks the centrality of slavery as the declared reason that states seceded. The state of Mississippi, for example, declared that its allegiance was with that of other slave states. South Carolina took a similar position, arguing that abolitionist sentiments in other states had threatened the institution of slavery. Such sentiments had, according to the state of South Carolina, led to insurrection on plantations. Free states had also allowed for the harboring of fugitive slaves, thereby "[disturbing] the peace and [concealing] the property of the citizens of other States."
The declaration reminds the union of the deal it had made in the Constitution to recognize each state as an equal with control over its own institutions. Furthermore, "The right of property in slaves was recognized by giving to free persons distinct political rights." In other words, the liberties of fellow citizens were dismissed in favor of antislavery sentiment, placing South Carolina and other Southern states at odds with nonslaveholding states.
https://avalon.law.yale.edu/19th_century/csa_scarsec.asp