He calls it "the end of The Batman," and even though he ends up being wrong, he's not entirely wrong. So when Gordon gets a call that The Riddler wants to see him at Arkham, he assumes his entire crime fighting career is about to come to an end. Based on his news clippings, Bruce is concerned that Nashton has figured out who The Batman really is. When rooting through his apartment for more clues about his motivations, they see that he has one last password-protected video to unlock. However, Falcone's demeanor here suggests that he really does feel that Thomas was asking him to kill and that despite his manipulative machinations, he thinks any other interpretation is a joke. Later, Alfred clarifies the nature of Thomas and Carmine's relationship, repairing the damage Falcone does to Bruce with his later revelations.
"Hippocratic" sounds to Falcone like "hypocrite." Even though Bruce thinks his father was a good man, Falcone remembers the day Thomas Wayne asked him to handle a problem for him that resulted in a murder, so he finds it utterly hilarious this overgrown boy doesn't know the full scope of his own father's complex morality. But just like all of The Riddler's clues throughout the film, it's a pun. Bruce doesn't understand what's so funny, and in the moment, neither do we. A soft chuckle at first, then something much heartier, down from the depths of his gullet. Falcone repeats his words to him and begins to laugh. Bruce says it means his father took the Hippocratic Oath, something every doctor must do. "You don't think that means something?" Falcone asks Bruce.