They identify 16 states as having violated their obligations (14 are listed as a group, and Ohio and New Jersey get mentioned elsewhere). These states had 176 electoral votes. Lincoln received 180 electoral votes, 173 from these states (3 of New Jersey’s went to Douglas), and 7 from California and Oregon.
The votes had not yet been officially counted when South Carolina pulled out, but the writing was on the wall and memories of what had happened very fresh.
Is it a union under a constitution or a union under something else?
So Lincoln won the way Trump won, and a lot of people were dissatisfied with that result, even though it was in accordance with the Constitution of the United States. The electoral college chooses the president, like or dislike the results. that is the law of the Land. So the Southern States were dissatisfied that the Nation followed the Constitution in electing Lincoln, instead of their man.