When South Carolina seceded from the Union in 1860, the main justification of secession was that southerners believed secession was the only way they could preserve their institution of slavery. Not all southerners supported secession, however, but those who did argued that because the southern states were a part of the Union, their rights should be respected by the federal government.
Because of this, many southerners believed that if the federal government was to abolish slavery, the prohibition would be an infringement on the states' rights and would therefore be unconstitutional. Southerners referred to the founding doctrine of the U.S., the Declaration of Independence. In it, the idea is presented that a country could change governments when the one in power was no longer right for the people.
The Declaration of Independence had originally been referring to Great Britain's rule on the colonies in 1776. In this new context, southerners justified their secession by claiming that forcing the southern states to abolish slavery was unlawful. They believed that this gave them precedent to leave the Union and start their own government, the Confederation, where their slavery institutions would not be abolished.
The decision to secede from the Union was not a unanimously favored one. In fact, some people in South Carolina disagreed with the decision to secede. Many of these southerners felt that the southern states were ill-equipped to form their own government. They also believed that secession would almost certainly lead to war with the Union, of which it did.