The primary reason that the southern states voted for secession was because they saw Lincoln as anti-slavery. Though he knew he did not have the legal power as of 1861 to eliminate slavery altogether, many southernors believed that by barring slavery in the territories, the anti-slavery states would be joined by the new anti-slavery territories when they became states, thus giving the anti-slavery forces enough political power to legally ban slavery everywhere. Lincoln was a symbol of the increasing political power of anti-slavery forces.
Plus, the rising anti-slavery tide in the North and presumably in the new territories would provide a progressively more welcoming escape route for southern slaves, since many northern states were turning a blind eye to recapturing slaves.
The south saw the handwriting on the wall on slavery, and seceded.