Remember that the Civil War was essentially about secession. The southern states left the Union and the northern states remaining fought to restore it. As historian Shelby Foote argues, the southern states never would have joined the country in the beginning if they didn't think they could leave. Therefore they believed it was a state's right to secede and the federal government could not stop them.
The fact that Lincoln had been elected in 1860 without a single southern electoral vote (his name wasn't on the ballot there) was the last straw, as they felt his abolitionist tendencies would trample on their entire way of life regarding slavery. So this was a war caused by the states vs. the federal government over issues of power.