From the Texas Letter of Secession:
"We hold as undeniable truths that the governments of the various States, and of the confederacy itself, were established exclusively by the white race, for themselves and their posterity; that the African race had no agency in their establishment; that they were rightfully held and regarded as an inferior and dependent race, and in that condition only could their existence in this country be rendered beneficial or tolerable.
That in this free government all white men are and of right ought to be entitled to equal civil and political rights; that the servitude of the African race, as existing in these States, is mutually beneficial to both bond and free, and is abundantly authorized and justified by the experience of mankind, and the revealed will of the Almighty Creator, as recognized by all Christian nations; while the destruction of the existing relations between the two races, as advocated by our sectional enemies, would bring inevitable calamities upon both and desolation upon the fifteen slave-holding states."
From the Mississippi Declaration of the Causes of Secession:
"In the momentous step which our State has taken of dissolving its connection with the government of which we so long formed a part, it is but just that we should declare the prominent reasons which have induced our course.
Our position is thoroughly identified with the institution of slavery-- the greatest material interest of the world."
From the Georgia Declaration of the Causes of Secession:
"The people of Georgia having dissolved their political connection with the Government of the United States of America, present to their confederates and the world the causes which have led to the separation. For the last ten years we have had numerous and serious causes of complaint against our non-slave-holding confederate States with reference to the subject of African slavery."
From the mouth of the Confederate States' Vice President, Alexander Stephens:
"Our new government is founded upon exactly the opposite idea; its foundations are laid, its corner- stone rests, upon the great truth that the negro is not equal to the white man; that slavery subordination to the superior race is his natural and normal condition. This, our new government, is the first, in the history of the world, based upon this great physical, philosophical, and moral truth."
As for the argument for states' rights, their complaint was highly hypocritical, as they argued against the northern states' rights to not return escaped slaves.
The South Carolina Declaration of Causes of Secession, (In which, slavery is referenced 18 times while states rights is only referenced twice) complains about non-slave states not obeying the federal laws requiring the return of escaped slaves.
"The Constitution of the United States, in its 4th Article, provides as follows:
“No person held to service or labor in one State, under the laws thereof, escaping into another, shall, in consequence of any law or regulation therein, be discharged from such service or labor, but shall be delivered up, on claim of the party to whom such service or labor may be due.”...
...But an increasing hostility on the part of the non-slaveholding States to the Institution of Slavery has led to a disregard of their obligations, and the laws of the general government have ceased to effect the objects of the Constitution. The States of Maine, New Hampshire, Vermont, Massachusetts, Connecticut, Rhode Island, New York, Pennsylvania, Illinois, Indiana, Michigan, Wisconsin and Iowa, have enacted laws which either nullify the Acts of Congress or render useless any attempt to execute them."
The rights of the states was not the real reason for their secession, and only added that when it suited them. I think the states' declarations speak for themselves. In the same way the Southern Baptist Convention split from the Baptists over slavery, so did the southern states split from the Union.