Literally, what has happened here is that the men have pumped out Mildred Montag's stomach and they have replaced her blood. They are leaving with her old blood and the contents of her stomach there in their "suitcase," as the one man calls it.
Bradbury calls this their case of liquid melancholy because they are taking away the stuff that (physically speaking) has been making Millie unhappy. This is a bunch of liquid that has caused her to be sad.
Of course, they are not really curing her. What they are doing is just temporary because the real cause of her melancholy is the fact that her life is empty.