Ike went from winning Missouri by 2% to losing it by 0.2%. The swing isn't that significant.
This is like asking "why did FDR win Michigan in 1944?" Yes, he had lost it four years earlier (and the Midwest generally swung right in 1944), but it's a matter of a 1-point win versus a 0.3-point loss.
That said, why did FDR lose several plains and midwestern states in 1940 and 1944? I have never seen any material covering anger towards FDR in this period. One would think places like Ohio and Michigan benefitted greatly from his policies.
Swings against FDR were especially sharp in areas where a large share of the population was of German ancestry.
Also, some people grew increasingly leery of the New Deal, and some felt that FDR had been President long enough, and there were other reasons.