“cheer, support,” 1889, American English, originally in a baseball context, probably from root (v.1) via intermediate sense of “study, work hard” (1856). Related: Rooted; rooting.
For root (v.1) they have:
“Associated with the verb sense of root (n.). Extended sense of “poke about, pry” first recorded 1831.”
I can see how you could get to studying from the idea of rooting about in books, but I don’t see the connection from studying to cheering and I’ve never heard the study sense of root. Did this “intermediate” sense have a short lifespan? Any ideas how we get from studying to cheering and why baseball?
They show the “modern sense” of the word “baseball” from 1845, so it would seem that rooting and baseball got together pretty quickly.