That's a real mystery. Illinois had been a solidly Republican state before that 1932-1948 stretch and its flip could be looked at as a return to form, but that doesn't explain the margins. Stevenson did terribly for a relatively popular incumbent governor, and a member of a distinguished Illinois political family no less. Nor did Eisenhower have some special appeal there- it was a Taft state in the Republican primaries. The statewide races were uneventful too, so no answers there.
My guess is McCarthyism-influenced backlash against the Chicago political machine. Theirs was the last big city machine left after James Farley nationalized the patronage system and immigrants rapidly became assimilated, prosperous, and no longer in need of extralegal aid in the 1940s, so it would have been a localized event. Cook County particularly was suburbanizing rapidly during that time with the availability of FHA and VA insured loans, the construction of thousands of single-family houses in Skokie and Oak Lawn, new expressways, and the move of many businesses to suburban locations. Also worth noting toward this theory is that Kefauver won the nearby Minnesota primary in 1956 by portraying Stevenson as a captive of corrupt Chicago political bosses.
Going slightly off-topic here, but Illinois wasn't
that Republican-leaning before 1932. Illinois was really a swing-state, if Republican leaning, during the Third Party System (1854-1896); it was one of the closest of the states where Lincoln won a majority (closer than Indiana, actually) in 1860, the state voted for Grover Cleveland in 1892, and John Peter Altgeld was notably elected governor during this period. During the Fourth Party System (Progressive Era/"System of 1896") Illinois was like New York State in that it alternated between being a Republican-leaning swing state and being just Republican enough not to be considered a swing state, but the instances in which Illinois was Republican enough to be considered safe R were often the result of Republican candidates being artificially boosted by Republican Chicagoland political machinery outcompeting and out-doing the Chicago Democratic machinery (helped by Catholic Church leaders denouncing the populist wing of the Dems and by Polish Catholics voting Republican as a reaction against rival Irish Catholic Democrat bloc voting) as well as boosted by the Socialist party(/ies) being disproportionately popular in urban Chicago relative to the nation at large and acting as a "spoiler" against the Democrats in Illinois (usually never enough to cause a Republican victory that wouldn't have happened anyway, but often enough to increase the GOP candidate's margin on paper by five or more points).
