FDR, FDR.
(The Democrats have always represented the cultural left, and the Republicans the cultural right).
Roots are with Bryan and his embrace of the Populist planks into his 1896 platform.
You could read the 1896 Democratic platform and then inform us of how left-wing you found it...
I said
roots. That does not mean the party dramatically lurched to the left, but began the process. And if you read the question, that is what the OP asked. If you take time to read the very link you posted, the roots are there:
Sure is government intervention in here.
In addition, Bimetallism was brought into the Democratic platform, to benefit the poor farmers.