It's slavery. I've written about it a lot, so I'm going to rerun prior response that gathers together previous work:

The short version is that there are two sovereign rights of the states which the slave states where keen on: 1) The absolute right to hold slaves in perpetuity themselves. 2) The right to suppress any antislavery movements or legislation that come out of the free states, generally through the force of the federal government. I've written about this from a few different angles before, so I'll take this chance to offer up my stuff as a companion to Zhukov's mega-post.

This is a high-level consideration of what state's rights theoretically means and the South's track record with it.

This discusses state's rights rhetoric during the Nullification Crisis, which is often cited as a case of "pure" state's rights unconnected with slavery despite being driven entirely by slavery. More on Nullification here.

This looks at the Confederate Constitution from a state's rights perspective, finding that in the ultimate moment the white Sough forged for itself an ironcland commitment to coercing slavery into states and territories should the need ever arise.

This has the usual quotations from the secessionists themselves on their motives.

The record is unequivocal, unless one takes a clear pattern of behavior (states rights when it will protect slavery, federal power when it will protect slavery, and most everyone switches sides right on cue) to mean the opposite of how it looks to any fair observer. Or one makes a superficial reading of it that refuses to ask the question you did.

"State's rights" isn't really an answer to anything. You always have to ask states rights for what, as you have. That's ultimately because it's a method by which to attain policy goals, not the goal itself. This is something everyone knew at the time, whoever was waving the banner. For it to be otherwise essentially requires us to believe that the section hazarded a war and risked countless blood and treasure to make a point of constitutional esoterica. That would make for a remarkably contorted, contrived argument which I don't see any modern historian taking halfway seriously.