What's the meaning of root in this context?
You root for the favorite. You cry when they go away.
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Here, root means to "cheer for".
It was perhaps most famously used in the 1908 song "Take Me Out to the Ballgame" in phrase "Let me root, root, root for the home team. If they don't win it's a shame."
The usage may come from the older English word "rout" that describes a cow bellowing.
Interestingly, the word root has a sexual connotation in Australia.
Here root is a synonyn for cheer on.
Apparently it was originally U.S. slang, although it has spread since then. The OED gives this sense as:
The first citation is from 1889. Here are some of the later ones:
Searching ODO for "root for" yields this: