I have written on this subject in this sub before here.

The short answer is that Presidential term limits had been talked about going all the way back to the Andrew Jackson administration. The talk gained some momentum within the Republican Party during the Progressive era. But nothing was ever done until 1947 when the Republicans gained a majority in Congress.

Enough Democrats willingly went along with it because they ensured the amendment would grandfather in the current president (Truman) before voting in favor of it, and because Truman was advocating for a change to presidential succession. A new succession act was passed by the same Congress that passed the 22nd Amendment, and then that new succession act essentially became constitutional a little while later with the passage of the 25th Amendment.

(The Amendment was unnecessary because the Constitution already gave Congress the power to set up presidential succession however they wanted, but the Amendment makes it more difficult to change due to prevailing political winds, as it had been changed several times between the ratification of the Constitution and the passage of the 25the Amendment.)

So the Republicans basically traded some Democratic support for the 22nd Amendment, which was in reaction to FDR's third and fourth terms, in exchange for Republican support for putting the Speaker of the House and Senate pro tempore next in line of succession for POTUS after VP, at a time when the VP position was vacant. Truman believed that elected officers should have priority over Presidential appointees.