I can't find the following phrase in an dictionary: "to throw down" [with somebody]. What does it mean?
"To throw down" means to fight someone (usually physical). It's a colloquial expression.
Interestingly, one entry on Urban Dictionary attributes its origin to "throwing down the gauntlet" an old tradition of throwing down ones gloves to challenge someone. The person challenging the other would throw down his glove or gauntlet, and the person accepting the challenge 'would take it up'.
You also see it used in popular culture like the TV show Throwdown with Bobby Flay, which is a show where Bobby Flay challenges people to cooking competitions.