How do you have insanity points influence the players?
According to the rules a character can have up to 39 insanity points (and provided they make their willpower roll successfully each time they test for mental trauma) be as sane as the newly minted character with no insanity points.
Ideally the players will roleplay some effects of increasingly large of insanity points but I’m tempted to also describe a scene and the NPCs people are interacting with differently depending on just how firm the PCs grip on reality is.
If anyone has ever played the 2004 game Vampire: The Masquerade – Bloodlines one of the bloodlines you could choose was the Malkavian which were somewhat less than sane. This completely changed how the game worked, newsreaders on the TV would address your character directly and it was possible to hold a conversation with a stop sign “No you stop!”
Obviously this would be difficult in a game where different characters would have different mindsets and different levels of sanity but it could also be really enjoyable.