The public was largely in favor of his third and fourth term runs. He won both elections in a landslide. The Republican Party tried to make a campaign issue out of it in both elections, with term limits part of their party's platform. It didn't work in preventing FDR's re-elections.

After his death, the Republican Party gained majorities in Congress in the 1946 midterms and made presidential term limits a policy goal. Enough Democrats went along with it to get the Constitutional amendment passed, in part because President Harry Truman wanted a change to the Presidential Succession Act, so it was a bit of give and take in passing the two laws. More information about the history of the 22nd Amendment can be found here.

Further reading:

• This post details the circumstances of George Washington's decision not to run for a third term. Summary: it's arguable that he intended to set any kind of precedent. It was Jefferson who was explicit about presidents not serving more than two terms.

• This post details how strictly the precedent was actually upheld. Summary: Not very strictly. Several two-term presidents before FDR had explored the possibility of a third term. Few two-term presidents who survived to the end of their second term hadn't explored the possibility.