Good questions, friend. I can quote from a few primary sources.

George Fitzhugh, in his 1854 broadside Sociology for the South: Or, The Failure of Free Society posited that the "peculiar institution" protected the weak and defenseless in ways free-soil and free-market capitalism utterly failed to do in "free societies" such as the North. "In free societies none but the selfish virtues are in repute", Fitzhugh said, "because none other help a man in the race of competition." Slavery, on the other hand, encouraged masters to protect those under him--his wife, his children, and of course his slaves, who Fitzhugh claimed would be subject to extermination in a "survival of the fittest" environment without the protection of their owners. "Slavery", Fitzhugh asserted, "protects the infants, the aged, and the sick; nay, takes far better care of them than of the healthy, the middle-aged, and the strong. They are part of the family, and self-interest and domestic affection combine to shelter, shield, and foster them. A man loves not only his horses and cattle, which are useful to him, but he loves his dog, which is of no use. He loves them because they are his" (1).

Now, even though Fitzhugh was an extremely prolific writer (aside from *Sociology for the South and his other well-known work, Cannibals All, he published tons of articles in the Southern periodical, De Bow's Review), he was probably the most enthusiastic about the "pro-family" aspects of slavery than anyone else. An equally well known and perhaps less esoteric defense of slavery can be found in congressman James Henry Hammond's "Mudsill Speech," as well as his two letters to Clarkson, an abolitionist. To briefly summarize both: In the Mudsill Speech, he told the Senate that the ruling class must be served by a lower, servile one--a social mudsill, so to speak--so that it could spend all of its time on ruling and other intellectual duties rather than physical drudgery, and also that black slaves were happier and healthier than poor freemen anyways. In his letters to Thomas Clarkson, Hammond claimed not to defend slavery in the "abstract," but that:

A: the God of the Bible allowed the "Hebrews" to purchase "heathen" bondsmen "forever" and that even Paul the Apostle returned slaves to their master (implying that whites were allowed to do the same for blacks and that slavery had divine sanction)

B: The black slaves, as a laboring class, were far less likely to rebel than poor whites in the North

C: Increased representation of the South due to its slaves was offset by increased federal taxation, and the North profited more from slavery than the South did

D: The free states had more debt, more "religious infidelity," and more mob and interpersonal violence than the slave states, implying slavery led to more social order, religiosity, and financial stability

and E: The numbers, demographics, and locations of slaves in the South made freeing them and paying them/replacing them with free labor impractical and expensive. (2)

Now for some secondary sources! The curious thing is, the robustly positive defenses of slavery Fitzhugh and Hammond gave (as examples) have a complex history and historiography. In the early 20th century, a historian by the name of William Sumner Jenkins claimed that slavery was only grudgingly tolerated by the South, until events such as the rise and profitability of cotton thanks to the cotton mill, fear brought about by the Denmark Vasey slave rebellion, and anxiety over abolitionist propaganda convinced many Southerners they needed to make a positive defense of the institution. More recently, however, Larry Tise has found that positive defenses of slavery go back as far as the 1700s, and that proslavery folks in the United States made pretty much the same arguments they did in, say, the West Indies. He also makes the point that Northerners defended slavery as well as Southerners, which strikes me as now well-established; look at the people in the North during the Civil War who "didn't want to fight for a Negro's freedom," so to speak.(3) But his book also has some more good anecdotes that illustrate why people defended slavery, and how:

In 1701, a guy named John Saffin got into an argument with a fellow named Thomas Sewall in Massachusetts (the north!) over the morality of slavery. Saffin argued that since slavery existed in the Bible, there was nothing wrong with him keeping slaves, though he grudgingly admitted that it was unlawful "for Christians to Buy and Sell one another for slaves." He also insulted blacks and said they were fit only for slavery. (4)

In 1748 George Whitefield expressed a concern for the well-being and religious instruction of slaves, but in practical terms, claimed Georgia "never can or will be a flourishing province without Negroes are allowed." (5)

According to Tise, the real change in Southern proslavery thought in the 1820s was when they started appealing to political economy rather than religion or "rights of man" (which blacks didn't deserve, they said). He quotes Zephaniah Kingsley from Florida who made in 1828 many of the same points Hammond did: In his treatise, Kingsley said that slaves were happier than "the common class of whites denominated as free." (6)

So, to sum up, some of the justifications of slavery in the U.S were religious (slavery is good because God allowed it in the Bible), social or economic (slave labor led to less social unrest than free labor, slaves were treated better than poor free whites, etc), and in Fitzhugh's case, moral (slavery protected blacks from being exterminated because it kept them under the loving, paternal hand of white masters).

Those are just a few examples, you really should check out all of Tise's book if you're interested, as he also analyses some of the most famous proponents of slavery, such as Stringfellow, Dew, and Dabney, I didn't have time to get into here. You should also read some of the books he cites, like Winthrop Jordon's White Over Black, Edmund Morgan's American Slavery, American Freedom, and Drew Gilpin Faust's A Sacred Circle: The Dilemma of the Intellectual in the Old South, 1840-1860 for even more background on the intellectual context of proslavery. I hope this helps! :D

1: George Fitzhugh, Sociology for the South, or the Failure of Free Society (Richmond, VA: A. Morris Publishers, 1854), p. 24, 46. Access it for free here: https://books.google.com/books?id=ITRcAAAAcAAJ&printsec=frontcover&dq=sociology+for+the+south&hl=en&sa=X&ved=0ahUKEwi3nLfioebJAhUEPT4KHaD8APIQ6AEIJjAA#v=onepage&q=sociology%20for%20the%20south&f=false

2: Mudsill Speech: http://www.pbs.org/wgbh/aia/part4/4h3439t.html

Letters to an English Abolitionist: https://books.google.com/books?id=Il1jAAAAcAAJ&pg=PA1&dq=hammond+clarkson&hl=en&sa=X&ved=0ahUKEwiDmvDypObJAhXDaz4KHSFYA_QQ6AEIKjAA#v=onepage&q=hammond%20clarkson&f=false

3: Larry Tise, Proslavery: A History of the Defense of Slavery in America, 1701-1840 (University of George Press, 2004), 1-10.

4: Ibid., 17.

5: Ibid., 21.

6: Ibid., 66-67.

There are several editions of American Slavery, American Freedom, White Over Black, and A Sacred Circle; I would recommend typing those names into amazon.com and picking up the most recent copies or the most affordable one, whichever is most important to you. Again, hope this helps :D