In short, the southern states reacted to Abraham Lincoln's election by seceding from the Union. A number of the southern states had already threatened to secede if Lincoln was elected. By the time Lincoln was sworn into office in March seven states had already carried through with this threat, forming the Confederate States of America under the leadership of their appointed president, Jefferson Davis. Four more states would soon follow suit. The next month, the war began with the siege of Fort Sumter in South Carolina. This signaled the beginning of open rebellion in the southern states.
While Lincoln's election was the catalyst for the secession of the southern states, it is quite possible that such a national crisis would have eventually happened anyway. The long series of compromises over the legality and proliferation of slavery had delayed the conflict for at least a generation but clearly failed to prevent it.
However, Lincoln had made it clear during his campaign that he opposed the spread of the institution of slavery. The southern states, fearing that the new president would make moves to outlaw slavery, or at the very least stop its spread, rebelled against the Union in an effort to create a territory in which the institution of slavery would be guaranteed.
It should be noted, however, that not all slave states responded to Lincoln's election by seceding. Kentucky, Maryland, Missouri, and Delaware remained loyal to the Union despite having slaves. Also, the western region of Virginia chose not to rebel and stayed in the Union which led to the creation of the state of West Virginia.