In the years before the Civil War (sometime around the 1840s, I think), the Baptist church in the south split over the issue of slavery. The new Southern Baptist faction was opposed to abolition and was also opposed to civil rights for blacks (even if they were freemen), so you can see how it might be fairly popular in the South, given the beliefs of most Southerners at that time.
If you compare that to other churches, the Methodists for example were also split over the issue of slavery, with northern Methodists strongly opposed to it and southern Methodists being much more tolerant of it. However, unlike the Baptists, the southern Methodists were not as clearly supportive of slavery. Southern Baptist clergy were free to own slaves. Southern Methodist clergy were not supposed to own slaves (though some did). Most other protestant Christian religions in the south held similar views (i.e. the official position of the church was that they tolerated slavery but did not recommend it).
The Catholic church didn’t split, but while the Pope vocally opposed slavery, church leaders in the U.S. were much more ambivalent about the issue.
If you are a white southern racist in the 1800s, are you going to feel more welcome in the church that openly supports your racism and slavery, or are you going to feel more welcome in the church that only tolerates it and recommends against it, or only locally supports slavery while the church’s leadership opposes it? You are going to go to the church that openly supports racism and slavery, of course.
Quakers were strongly opposed to slavery, and were largely responsible for getting the whole abolition movement going in the North. Their strong anti-slavery stance was probably the strongest factor preventing the Quaker religion from spreading into the South.
Many black slaves were introduced to Christianity through the Baptist church. Many evangelicals thought that it was their duty to bring Christianity to the slaves. Note that both pro-slave and anti-slave factions of the Baptists both used various Bible passages to support their claims. In most areas of the South, blacks were not allowed to lead churches, so blacks had to go to white churches (they were segregated, of course, with pews specifically for whites only and other pews set aside for blacks). After the Civil War, blacks were finally allowed to lead their own churches, so they split off from the white churches and formed their own branches of the Baptist church.
I don’t know enough about the Baptist church to say why they remained popular through the 20th century and to modern times, but the Baptist church certainly got a large boost during the 19th century due to the divisive issue of slavery and the fact that the church split into factions that strongly supported that faction’s beliefs.