Slavery was a major factor in the South seceding from the Union. It wasn't the only factor but it was a major driver. Lincoln before being elected already had announced that he intended any new state to be a "free" state. That was going to tip the balance to the non-slave holding states. The election of 1960 favored Republicans, which the South perceived as threats to their establishment.
Slavery was a way of life and the economic engine for the South (for the country in general, but specifically for the South). They weren't comfortable in what they were seeing from the U.S. Government and felt the momentum was going against them. There were some other issues as well, not saying it was all slavery, but they fought to preserve their way of life and slavery was a big part of it.
Now, if we are talking about the North, they didn't give a rats ass about slaves. They just wanted to preserve the union (i.e. resources/manpower/tax base).