"Put out" = provide sex (in return for dinner)... the joke here is the gender reversal: Phoebe is being the predatory one. In this case, "won't" has the meaning of "doesn't want to"/"refuses to".
As a phrasal verb, "put out" can have several meanings. As well as the above example:
Put outside = "I put out the cat last night."
Displease = "He was put out when he realised he had to work this weekend."