Voter suppression, especially of Black voters. Due to how racially polarized party support in the 'Solid South' was, these states were practically one-party states for the Dems due to it, all the way from the end of the Reconstruction era in the 19th century until Thurmond's candidacy in 1948.
Basically this. And IMHO even more crazy than 1936 is 1904 - TR won in a landslide, but in southern states (particularly MS/SC), the results were overwhelmingly (I'm talking 90%+) for Parker. In 1936, FDR still won in a landslide. Parker
lost in a landslide and still got ridiculous margins out of the deep south (and if you compare results in SC/MS to national results, you'll find the disparity was much bigger in 1904 than in 1936).