"The 22nd Amendment... has frequently been described as a reaction to the presidency of Franklin Roosevelt," wrote Thomas H. Neale of the Congressional Research Service of the Library of Congress in the monograph, "Presidential and Vice Presidential Terms and Tenure" (2001). "The amendment was a top priority for the 80th Congress (1947-1948), the first to be controlled by Republicans since 1931. Debate on the proposal was the occasion of considerable rancor, as some Democrats claimed it was both undemocratic and an act of posthumous revenge against Roosevelt, while Republicans argued their goal was the prevention of excess concentration of power in the hands of one person. The idea of term limits was not new, however: one scholar notes that 270 amendments to limit presidential tenure had been proposed prior to 1947. Moreover, the measure passed both houses by wide margins, and with some degree of Democratic support."
And, it should be added, three-fourths of the states ratified it over about four years. It's possible that Congressional ire at FDR played a part in the amendment, but it's unlikely that all those state legislatures shared it. Still, the amendment doesn't sit well in some quarters. "Proposals for repeal of the 22nd Amendment have been advanced on several occasions since its ratification," according to Neale. "These proposals have usually been offered during the second term of a President who has enjoyed a degree of success or popularity. Advocates of repeal assert that a popular or successful two-term President should be able to serve additional terms... [however] none of these proposals received any action beyond pro forma committee referral."