To be “rooting for someone” means that you support them and you will be cheering them on,
“To root for someone” is an expression we use when we want to let someone know that we want them to win. It is often used in sports but it could also be used in the context of life in general.
In my experience, this expression is most common in American English. Be careful though, It has a different meaning in Australian slang(to have sex).
A root usually means the foundation or support of something so for this reason it makes sense in this context too.
You can beat the cowboys, we are all rooting for you here.
You go get em champ, I am rooting for you
Other ways to say “I am rooting for you”
I support you.
I got your back.
Go get ’em!
You can do it!