Well, it was about "states' rights". It's just that the "right" in question was the one to keep humans as property.

From South Carolina's Declaration of Secession, the first state to secede:

A geographical line has been drawn across the Union, and all the States north of that line have united in the election of a man to the high office of President of the United States, whose opinions and purposes are hostile to slavery. He is to be entrusted with the administration of the common Government, because he has declared that that "Government cannot endure permanently half slave, half free," and that the public mind must rest in the belief that slavery is in the course of ultimate extinction.

This sectional combination for the submersion of the Constitution, has been aided in some of the States by elevating to citizenship, persons who, by the supreme law of the land, are incapable of becoming citizens; and their votes have been used to inaugurate a new policy, hostile to the South, and destructive of its beliefs and safety.

From Mississippi, the second state to secede:

Our position is thoroughly identified with the institution of slavery-- the greatest material interest of the world. Its labor supplies the product which constitutes by far the largest and most important portions of commerce of the earth. These products are peculiar to the climate verging on the tropical regions, and by an imperious law of nature, none but the black race can bear exposure to the tropical sun. These products have become necessities of the world, and a blow at slavery is a blow at commerce and civilization. That blow has been long aimed at the institution, and was at the point of reaching its consummation. There was no choice left us but submission to the mandates of abolition, or a dissolution of the Union, whose principles had been subverted to work out our ruin. ...

Utter subjugation awaits us in the Union, if we should consent longer to remain in it. It is not a matter of choice, but of necessity. We must either submit to degradation, and to the loss of property worth four billions of money, or we must secede from the Union framed by our fathers, to secure this as well as every other species of property. For far less cause than this, our fathers separated from the Crown of England.

Alabama's 1861 Constitution has a whole section about slavery, reproduced here:

Section 1. No slave in this State shall be emancipated by any act done to take effect in this State, or any other country.

Section 2. The humane treatment of slaves shall be secured by law.

Section 3. Laws may be enacted to prohibit the introduction into this State, of slaves who have committed high crimes in other States or territories, and to regulate or prevent the introduction of slaves into this State as merchandise.

Section 4. In the prosecution of slaves for crimes, of a higher grade than petit larceny, the General Assembly shall have no power to deprive them of an impartial trial by a petiti jury.

Section 5. Any person who shall maliciously dismember or deprive a slave of life, shall suffer such punishment as would be inflicted in case the like offense had been committed on a free white person, and on the like proof, except in case of insurrection of such slave.

The way most people claim "states' rights were the cause of the civil war" is as an alternative to the "slavery was the cause of the civil war" explanation. This is entirely incorrect. However, I think that it can be accurate, so long as the "states' right" in question is the right to have a legal system that treats humans as property. After all, that is what many of the people involved said.

Yes, division and dissention within the union occurred for other reasons. The economic split -agrarian south vs. industrializing north- was a real and persistent internal division that put real centrifugal force on the union. So was the changing cultural character of various states due to non-uniform levels of immigration, and political battles over financial, legislative, and judicial priorities. Regardless, many of the seceding states themselves identified their own positions with the institution of slavery. I have yet to see a plausible reason why they would do so falsely. It wouldn't give them domestic or international popularity, and wouldn't serve to 'paper-over' an uglier reason. The most parsimonious reason why they identified their political causes with slavery is because that's what their real political cause was.

This question is listed in the FAQ.

To a limited extent, yes arguably. Going to copy a comment I made on /r/badhistory since I am at work

In early January 1861 the Virginia house passed resolutions committing themselves to resist the coercion of any seceding state. On February 4th 1861 Virginians voted in favor of any action by the state convention must be submitted to the people 100,000 to 45,000. Secessionists opposed the popular referendum, and it was seen as a case test. Furthermore only 1/5 of the elected delegates were for immediate secession. Newspapers in Virginia routinely attacked the secessionists, the Charlottesville Review stated on January 4th that "We entertain towards South Carolina the most bitter resentment". Comments such as these were common throughout the Upper South with the Wilmington Daily Herald stating in November of 1860 that "There are no two adjoining states in the Union whose people have so little community of feeling as North and South Carolina".

In Tennessee on February 9th voters rejected calling a convention 69,000 to 57,000. Even more discouraging for secessionists had the convention met Unionists candidates enjoyed more than three times as many votes as secessionists, North Carolina would have had similar results with about three times the number of Unionists elected as secessionists. However on January 20th, Tennessee received word that New York had made an offer to raise troops to coerce the seceding states back into the Union. Outraged, Tennessee's government passed a resolution condemning New York's offer and stated

"indication of a purpose upon the part of the people of that State to further complicate existing difficulties by forcing the people of the South to the extremity of submission or resistance; and . . . whenever the authorities of the state shall send armed forces to the South for the purpose indicated in said resolutions, the people of Tennessee, uniting with their brethren of the South, will, as one man, resist such invasion of the soil of the South at all hazards and to the last extremity".

While no formal declaration was made, numerous public statements were made to the same effect in Kentucky and North Carolina and were not not challenged. Thus while the upper south rejected secession they nonetheless resolved that they would not abide any federal interference in those states who had already left the union.

To quote David Potter in The Impending Crisis: "Although the people of the South were badly divided on the issue of secession, they remained united in their belief that southern rights must be maintained and that no southern state could acquiesce in the use of federal force against any other southern state.