It is easy to answer this in simple political terms, much harder to answer it in moral/psychological terms.
Politically, the South seceded because Lincoln was elected and the South felt that was the last straw. The South felt that the North had been systematically abusing its rights and running the country solely for the benefit of the North. They felt that the North was bent on imposing its will (mostly on the issue of slavery) on the South and that Lincoln's election (with no Southern support) meant that the "oppression" was going to continue and to get worse.
Lincoln went to war because he believed so strongly in the sanctity of the Constitution and the Union. He believed that the Constitution was a sacred pact between the people of the US and that it must not be torn apart.
But I think your question asks something more. Why would people go to war over such issues as this? It really seems that the true cause of both of these things was overweening pride. For two sides to spill blood rather than to give in over issues like this shows the strength of human pride. It shows that people tend to be willing to go to extreme lengths to have their own way and to be "proven" right.