1
general::
Cheer on, give moral support to, as in The fans were out rooting for their team, or I've been rooting for you to get that promotion. This expression may come from the British verb rout, which is used of cattle and means "bellow." [Late 1800s]
American Heritage Idioms
2
general::
Phrase(s): root for someone or something
to cheer and encourage someone or something. • Are you rooting for anyone in particular, or are you just shouting because you’re excited? • I’m rooting for the home team.
McGrawhill's American Idioms And Phrasal Verbs