So I will give the official reason why Snow laughed,
First of all: Read or Watch the part where they were talking in the gardens (Towards The End) Now snow talked to Katniss and basically told her that "Your sister is dead because of Coin, and even though she was not aiming for your sister, she was aiming for the Capitol's children, regardless no matter how bad I was (Snow) I would never be wasteful (Do things for no reason) or kill Children (Being a Monster).
Now, no matter what you say about Snow, he was an honorable man, even if he was over the board crazy, he gave katniss this information to show he was not the person she really thought he was, but rather a man simply doing whatever he could to keep his power.
Second of all: Read or Watch the part where Coin and the rest was in a meeting discussing who was in charge and what was going to happen next (Sometime after the Garden Scene). This part was basically all Katniss needed to see to prove who the real enemy was, in her point of view (Remember what Snow said and explained to her, Coin was indeed after the Capitol's children YES Coin was also trying to take Snow's position in command YES Coin's last straw that proved she had to go was when she talked about how the people could't decide for themselves).
Even though it was the people that helped get this freedom, and showed everyone else what they can do together, Now this probably hit Katniss as she left that meeting.
Third of all: The part where she entered the arena (Immediately after the meeting Scene). This part basically was about fueling revenge for Katniss on the (So thought to be Rebellious Commander Coin). When Coin gave her final Speech she would ever say, she would have never known, she was talking about the injustice within herself (This is the reason why snow laughs), because it's ironic how everything she said fits her description of evil and twisted no better than himself, he also laughs because after everything he put Katniss through, she still had the will to trust him, making him feel great knowing Katniss did it and took down the real enemy of it all.
And this is why Snow laughed, these 3 scenes will show why he laughed, and who was really in charge the whole time!
