1
general::
Phrase(s): cheer someone or something on
to encourage someone or a group to continue to do well, as by cheering. • We cheered them on, and they won. • We cheered on the team. • Sam cheered Jane on.
McGrawhill's American Idioms And Phrasal Verbs
2
general::
Encourage, as in The crowd was cheering on all the marathon runners. Originating in the 1400s simply as cheer, this usage was augmented by on in the early 1800s.
American Heritage Idioms