How do the Silent Assassin requirements work

Killing or subduing an enemy will result in a loss of points. A loss of points means it takes longer for you to unlock the improved abilities that 47 can gain and making it harder to gain the coveted "Silent Assassin" rating. However, depending on how you perform the kill and take care of the situation, you will gain points back. If executed correctly, you can end up with a net zero point change even after killing someone. For example, if you perform a silent kill (eg. fiber wire with nobody else looking), you will lose a set amount of points for killing someone, but then gain half of it back for doing the silent kill. Then, if you hide the body, you will gain the remaining points that were lost, resulting in no change. This also works for subduing people, then hiding them.

All in all, this method allows you to still earn a "Silent Assassin" rating while still being able to deal with certain people that are directly in your path.

Also, killing or subduing someone will gain you a set amount of instinct, so if you're running low and you depend on instinct, this is a viable way to get more. I'm not sure how instinct works on Purist difficulty.

Edit: It seems that there is no instinct on Purist difficulty, at least in the sense that you can't gain more. On Purist mode, trying to use instinct will only show you where your current target is and slow time down a bit, nothing more. This ability also appears to be unlimited.

A usual and practical way to restore Instinct is by finding a nearby NPC to kill/knockout silently and then hiding their bodies in a container. Check if there's a container nearby where you can safely hide a body before going in for the kill/knockout. Make sure the kill/knockout is silent.

You can also gain Instinct by "hiding". This can be done by pressing the "hide" button when a prompt comes up near an object where you can hide.
^{(Thanks to Ben for mentioning.)}
If you are in a position where you can collect evidence or achieve a mission objective, then you might want to go for those instead, as doing those will also restore Instinct.

To prevent Instinct from going down, avoid being noticed by NPCs wearing the same clothes as your disguise. Sneak/crouch whenever possible when around these NPCs; this will lessen the need for instinct mode.