I think it means "support themselves" also.
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User_garyand we told the guys to back themselves,I've never heard the expression in the US.
Mr WordyI also believe it means "support themselves", or "be confident of their own success".I might agree with the second.
Avangi In the US, we just don't use it that way.It's the same in BrE too. "back" is often used to mean "bet on", so "told the guys to back themselves" could mean "bet on themselves". I just assumed that in
We might say, "Who are you backing for the finals?"
It primarily means, "Who are you betting on?"
But it could mean "Who are you cheering for?"