The Republican party arose out of a growing abolishionist sentiment in the north, although they generally took a long view of taking care of slavery (prevent its spread to new states, then gradually free slaves while financially compensating their owners, shit like that, rather than just up and abolishing it in one day). Southerners generally looked at Republicans as an insane fringe group (think a liberal Tea Party), and so started increasingly freaking out that they were actually gaining traction in the north. This hit its peak when Lincoln was elected, as the south felt that if a Republican could be elected president, it was only at matter of time before the government kicked down their doors and stole their slaves.
Of course, only the upper class had slaves in any real numbers, but the lower class still considered slavery part of their culture, and saw "slave" as the proper station for blacks; freeing blacks would theoretically put blacks on the same level as lower-class whites, which couldn't be stood.
The south decided to preempt what they saw as the now-inevitable move to abolish slavery by seceding for the US. The language for seeing the issue of slavery as a matter for states to decide had already been fairly well established by this point, so the southern states used the violation of that states' right as the basis for secession.
Of course, the US had military bases located in the south, which were still occupied by the US Army. Confederate troops eventually managed to forcefully push northern troops out of Fort Sumpter, and that attack on a US fort served as the official reason for declaring war. In the north, the rhetoric initially focused on preventing what was seen as an illegal secession from the US, with slavery coming into it somewhat later.
So, to sum up, the south fought the Civil War to preserve slavery, while the north fought to prevent secession and then thought to themselves, "you know, as long as we're here..."
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