Deep Roots
First, relative political conservatism and liberalism are remarkably stable in any given geographic area over time (even though, in absolute terms corresponding to specific policies, almost every place gets more liberal over time).
For example, even at a county by county level the 1876 Presidential election and the 1976 Presidential election look almost identical (flipping the Democratic and Republican parties, which traded places on the political spectrum in that time period). The fact that the Democratic Party and Republican Party traded places is weird, but largely just a branding issue that hides long term ideological stability.
At the state level, the relative positions of representatives of U.S. states in Congress on issues like national security, defense spending, foreign policy and social issues on a liberal to conservative scale from the 1790s are essentially the same as they are today in most cases.
Incidentally, this isn't unique to the U.S. You see ideological stability over similar time frames all over the world, for example, in regions within Germany (e.g. the modern politics of Saxony with deep historical roots and the continuing Protestant and Catholic divide in a far more secular country), Italy (e.g., between the North and the South were were united only in the 1870s), and the U.K. (e.g., differences in political attitudes and institutions between Scotland and England) associated with early modern kingdoms that predate these modern states and had not yet been fully consolidated. Outside the West, regional political differences in India, and differences in attitudes and political culture between Northern China and Southern China similarly have deep roots, as do those in Nigeria which is practically two different countries with an Islamic North and Christian/Animist South at the moment).
So, the bottom line is that the highly conservative leanings of the American South are largely legacies of the pre-Civil War slavery regime. In that environment, most of the South had free whites and lots of black slaves. The distrust between the two ran deep.
This reality is explored much more deeply, for example, by David Hackett Fischer in Albion's Seed, and by Colin Woodard in American Nations: A History of Eleven Rival Regional Cultures of North America. Also useful in understanding this sweep of American political history is Randall B. Riley, in his book, Congress: Process and Policy, the first half of which illustrates how much the federal government and its institutions were transformed from its pre-Civil War incarnations by the U.S. Civil War, and then, by the New Deal and World War II.
Little Diversity Or Disruption Due To Migration
Secondarily important is that from the late colonial era until the 1980s or so, the American South had much less immigration from outside the U.S. and from elsewhere in the U.S. than other regions of the country. So there was no one to shake up the mix very much.
This is because immigrants migrate to places that are the most economically prosperous, and that wasn't the South for most of its history.
This also means that whites in the American South, until the last 40 years or so, were extremely homogeneous culturally and in sources of ancestry.
At that time there were very few Catholics outside of Florida, Louisiana and Texas, there were almost no Lutherans, and there were very few Presbyterians or Congregationalists or Unitarians or Universalists or Jews or Quakers or Muslims or Mormons. Baptists, Episcopalians (restricted to a thin upper class), Methodists, and black Christian denominations with a shared but parallel theological origin, were pretty much it - although some rebranded as non-denominational or into splinter sects.
Some of this religious homogeneity was also because it was freshly emerged. Before the Second Great Awakening (ca. 1790 to 1840), the American South was the least religious part of the United States. This flipped at this time, with a new uniquely American style of faith emerging all at once and securing mass conversion to active, fervent Christianity. This new version of Christianity fit prevailing social attitudes at the time in the South to a "t". The Second Great Awakening "ironed out" many small ideological and cultural differences that had been present in the South between people there, before it actively embarked on uniting these folks in a common ideology and faith.
There was no significant linguistically different immigrant population and little linguistic diversity period outside old French in Louisiana and legacy Mexicans in parts of Texas. (Fun Fact: the Spanish language spoken by people native to New Mexico and Southern Colorado retains many words and language constructions from Colonial era Mexico that are no longer found anywhere else in the world.)
There were few Latinos (outside Florida and Texas). There were almost no Asians. The Native Americans who were there at the start were mostly dead, or exiled, by the time the Trail of Tears (1830-1850) was concluded.
The presence of only two very disparate cultural groups that are internally relatively homogeneous (i.e. white Southerners and black Southerners) leads to polarization.
A Narrower Range Of White Affluence Than The North
A tertiary factor is that despite our image of the South as dominated by big plantations epitomized by a home grown de facto aristocracy of plantation owners v. working class whites and black slaves in the Antebellum era, this is somewhat misleading. While there were lots of big plantations with wealthy families in them, there were almost no families that had wealth on the scale of Northern manufacturing, rail, shipping, and finance based businesses that were far more highly concentrated in wealth. The bottom also lacked the layer of dirt poor, just-the-shirt-on-their-backs new immigrants that was found among Northern whites.
During Reconstruction, many of the wealthy planter families experienced permanent or temporary total financial collapse, in part, at the hands of Northern carpetbagger financiers.
About 18% of white men in the South aged 13 to 43 died in the American Civil War (1861-1865) and Reconstruction, and many more were crippled as a result, for example, with amputated limbs.
The war destroyed much of the wealth that had existed in the South.
All accumulated investment in Confederate bonds was forfeit. Income
per person in the South dropped to less than 40% than that of the
North, a condition which lasted until well into the 20th century.
Southern influence in the US federal government, previously
considerable, was greatly diminished until the latter half of the 20th
century.
An effective Union naval blockade captured about 95% of the exports
from the Confederate states during the war. In the short run, the
war destroyed almost all 8,800 miles of Confederate railroads while
the Union added about 7,300 miles to its existing 21,800 miles of
railroad. The war also destroyed almost all of the South's
manufacturing plants, and almost all of its cotton production, and
almost all of its exports (70% of the total for the nation before the
war).
(The quote is a second hand Wikipedia quote to which my link has rotted.)
The political ideology impact of the post-Civil War economic collapse in the South can be compared to the radical right influence that economic collapse in Germany after World War I in the Weimar Republic had leading to to fascism and World War II.
So, Southern whites, coming out of Reconstruction, were far more economically homogeneous than elsewhere in the U.S., which helped build a sense of solidarity.
Also, in 1860, there were about 22,100,000 people living in the Union states (only 400,000 of whom were slaves, about 2% of its overall population, of whom 340,000 were in Kentucky and Missouri) and 9,100,000 living in states that would become a part of the Confederate States of America (about 3,500,000 of whom were slaves, about 38% of its overall population). In the lowlands deep South including the Mississippi River valley, the percentage of the population that consisted of slaves was much greater.
The transition from a society where white Protestant men made up 98% of voting population to one where blacks made up 20%-50% of the voting population in particular counties and other political jurisdictions encouraged tactics of voter suppression and political unity in the face of a "common enemy" that was much more different from them than the political differences that existed among white Southerners.
Poor, Less Educated, Economically Insecure People Trend Conservative
Over all of that time, from the late colonial era to the present, the South was and remains less affluent as a whole than the North and less economically secure.
This wasn't confined to the rich either. Flush toilets and electricity for ordinary people came many decades later to the South than it did to the North too.
Literacy in the South among whites was also profoundly lower than the rest of the U.S. until long after the Civil War. A litmus test of the distinction can be seen these statistics:
In 1850…Arkansas had 97,402 white persons under twenty, and only
11,050 attending school; while of 210,831 whites of that age in
Michigan, 112,175 were at school or college. Last year, Michigan had
132,234 scholars in her public common schools.
In 1850, Arkansas
contained 64,787 whites over twenty, – but 16,935 of these were unable
to read and white; while, out of 184,240 of that age in Michigan, only
8,281 were thus ignorant, – of these, 3009 were foreigns; while, of
the 16,935 illiterate persons of Arkansas, only 37 were born out of
that State. The Slave State had only 47,852 persons over twenty who
could read a word; while the free State had 175,959.
Michigan had 107,943 volumes in “libraries other than private,” and Arkansas 420 volumes.
From Theodore Parker’s The great battle between slavery and freedom (1856).
And, in the long run, on average and barring intervening factors, poor, economically insecure, less educated people tend to be more conservative.
Obliquely related is that the American South was to a much greater extent a zero-sum game economy than other parts of the U.S. The South has a mostly agricultural economy until very late compared to the rest of the U.S., and almost all arable land in the South was farmed very early on.
In contrast, the Midwest and then the West allowed for win-win solutions by expanding into new territory (and displacing the Native Americans there). And, the North and Midwest also overcame the zero-sum game trap by industrializing so that agricultural land wasn't a limiting factor, an economic trend that came really a century or so later to the South than to the North.
Conservative thinking and political approaches are a better fit to a world that is a zero-sum game. Liberal thinking and political approaches are a better fit to a world where win-win solutions are widely available.
It is the liberalism of poor non-whites in the U.S., and the liberalism of whites in unions post-WWII, and not the conservatism of other poor whites in the U.S., that is the historical exception that requires explanation.
To oversimplify, poor non-whites in the U.S. are liberal because conservatism was taken by poor whites whose policies posed an existential threat to them, and working class union whites were liberal because the economy needed so many modestly skilled workers after WWII that they enjoyed unprecedented prosperity that unions helped to harness with liberal policies.