The southern states were largely agrarian at that time, which made slavery profitable, while the north was more heavily industrialized, which didn't favor slavery. From the southern point of view, if slavery was abolished, it would wreck their economy. So they saw no choice but to secede.
I would point your attention to the Constitution to Article 1, Section. There you will find all delegated authority assigned to the federal government. What you will not find is any mention of secession. The union was an agreement among states, not a binding law.
The good people of the South had the Declaration of Independence to provide them guidance in making the monumental decision to leave the union:
"… that whenever any form of government becomes destructive of these ends, it is the right of the people to alter or to abolish it, and to institute new government …"
And that is what Dixie did.
"When Faith preaches Hate, Blessed are the Doubters." - Amin Maalouf