In a tennis match, for instance, between Roger Federer and Rafael Nadal, I would "root for" Nadal (ie, I want him to win and I would clap and cheer loudly for him).
It doesn't have to be sport-related though. If a friend of mine were going for a job interview, I would say "Good luck. I'll be rooting for you." I would simply mean that I will be mentally hoping that they get the job.
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