I heard the phrase "Take on your enemies" announced in a news conference today. Could you help me understand what it means?
To take on someone or something is to confront them.
If you take on your enemies, you confront them with the goal of defeating them.
Take on your enemies means - go, deal with them (bravely?).
It's used as a transitive verb.
Take on (tr. v) - to begin to perform or deal with
Note that there are other uses of it on the page as well.