To be fair to the author, he is saying that southern apologists for slavery used (their interpretation of) Burkean conservatism to justify slavery as a positive good. This is quite true.
However, opponents of slavery and particularly its moderate opponents like Lincoln, also appealed to Burkean conservatism, and IMO more appropriately.
He's saying that slaveowners and supporters of slavery used Burkean arguments in their defense of slavery. Then or now, people get arguments wherever they can find them when they try to defend what they take to be their vital material interests.
Lincoln and the Whig Party he belonged to were themselves conservative in very real ways, and when you consider all the talk in Charleston about the "Revolution of 1860 [or 1861]" it's clear that on can't really describe slavery, its supporters, and their actions wholly as Burkean or conservative developments.