Follup question: Today, I associate Episcopalians with being a particularly progressive denomination (for example on gay marriage). Was that not the case in the 1800s? Or would Episcopalians outside of the south have been more likely than others to be active in the antislavery movements?
To answer this, we need to go back to the beginning of the colonies. We all know that some of the early Americans fled England to escape religious persecution. This is true enough, but England did have an officially endorsed religion: Anglicanism, or the Church of England. Most estimates say at least 80% of England at least pretended to be Anglican and Anglicans quickly became a large presence in colonial society.
The English Civil War broke out in the 1640s and '50s. This had two effects. First, many Anglican gentlemen and nobles fled to the colonies, especially the slaveholding colonies where they could buy slaves and estates. These estates, along with the relative power of the Anglican Church there, allowed them to live as they had despite a Puritan England.
Secondly, after Puritanism fell in the Restoration, most of the colonies passed acts for religious toleration. However, nowhere had universal religious toleration until Pennsylvania was formed. There was only one group that was tolerated everywhere: Anglicans. Catholics were allowed in Maryland but not in Massachusetts, for example. Likewise, most royal officials would be Anglican and the English government, from time to time, supported Anglicanism in the colonies. This was sometimes resisted, most notably under the reign of King James II (who was a Catholic and unpopular and, fun fact, the namesake of New York).
Nevertheless, on the eve of the Revolution Anglicanism was a robust force in the colonies. It was the richest church, probably the most populous, and had the most state support, both from several local state governments in the south and from England.
But the Anglican Church in the US was loosely under the Bishop of London and held as one of its tenets that the King was the head of the church. Naturally enough, the Church was overwhelmingly loyalist. Only about 27% of the church structure were patriots, though this was much higher in some parts of the South. The Continental Congress criminalized praying for the King or Parliament in 1776 which resulted in almost every northern Anglican church shutting its door. After the Revolution, the Church was disestablished, much of its land seized, subject to fines, and individual clergy were investigated. Between 70% and 97% of Anglican clergy were expelled or fled, depending on the colony. By some reports, there were only two Anglican priests in the whole of New England after the Revolution.
Those who remained radically reformed the church to have more democratic structures and a more American structure. They revised the Book of Common Prayer and attempted to strengthen what churches remained. But their congregation's numbers were devastated by the Revolution, as was their influence. By 1790, no state officially supported Episcopalianism or Anglicanism anymore.
But it remained the religion of the elite, especially the Southern elite. The Church had been relatively more patriot in the South, it had definitely been more integrated into local society, and the South was a more elite dominated society. There was also less tradition of religious dissent and a longer tradition of conformism. Likewise, the First Great Awakening had been more a Northern than Southern phenomena, especially in the cities and coasts. Where it did find hold in the South, it was often in more marginal populations, like rural whites (such as those in Appalachia) or slave communities. The same was true of the Second and Third Great Awakenings, though these were more present in the south. Notably, these awakenings often stressed social equality and opposition to slavery, which made them unappealing to slaveholding elites. So while it declined in general, it remained strong in the South, especially along the coasts and in major slaveholding regions. (It was also still an elite religion in the North, but Northern elites were less dominant overall and were mostly not slaveholders.)
Between the Revolution and Civil War, Episcopalianism mostly focused on keeping the populations it had or expanding in the areas it already had strength. It was on the defensive for the first half of the period in the face of the loss of power and then the Second Great Awakening. When it managed to regain some followers later on, it was mostly among the classes and in areas where they already had widespread support. They did not successfully expand out west. They thus expanded their share among Eastern and Southern elites but continued to decrease overall. Even as the religion became smaller and smaller, some upwardly mobile people continued to convert in an attempt to assimilate into the upper classes.
In the 1840s and '50s there was the Slaveholding Bishops controversy. This is a very complex issue in of itself, but it can be summarized basically like this: bishops and the church owned property. In the South that often included slaves. If slavery is a necessary evil, the Church should not engage in it so as to keep it free from sin. But if slavery is good or neutral, then the Church should own slaves, since they are a great return on investment (if neutral) or improve the life of African Americans (if good). Episcopalianism didn't exactly endorse slavery, but they did allow bishops to own slaves where it was consistent with the law. Other churches did not, and those often split. (There was a parallel, and more direct debate, among the Low Churches which had similar effects).
Upon the start of the Civil War, any churches that remained united split (with the notable exception of the Catholics). This included the Episcopalians, whose Southern members seceded from the church.
While every religion had members on both sides, it is possible to roughly divide sympathies by religion. Evangelicals, Catholics, and most religious minorities (Jews, Mormons, etc) were disproportionately Unionists. Mainline Protestants were disproportionately Secessionists. For example, Lincoln was Baptist, Grant was a Methodist, and Sherman was a Catholic, while Davis and Lee were Episcopalians and Jackson was a Presbyterian. West Virginia was a stronghold of evangelical beliefs even back then and seceded from Virginia to stay in the Union. Among the 40% of the Virginian officer corps who remained loyal to the Union, non-mainline Protestants made up a disproportionate number of those who stayed. In eastern Tennessee, where Evangelicalism was relatively strong, the Confederacy had to resort to military occupation.
To be sure, much of this was cultural and regional. For example, West Virginia and Eastern Tennessee were both Appalachian and had many fewer slaveholders. And it should be emphasized that there were significant populations of both Evangelicals and Mainline Protestants on both sides. The Confederate camps, especially later in the war, were noted for their religious revivals. And the Northern Episcopalians were very supportive of the Union. While the Civil War had religious elements, it was not a religious war between sects. But religion did play a very important role in the politics of the day.
So, in summary, two reasons: Firstly, Episcopalianism was the dominant religion of the slaveholding Southern elite. Secondly, Episcopalianism was relatively friendly to slavery and Episcopalians more likely to support the south.
Sources
What other denominations were strong in the South at the time, or had enough of a presence to be generally accepted? I recall learning about Methodism and the Wesleys having a connection to my state (GA) in state history class.