Laying Low is a term that refers to a frequent tactic of the Mafia and Evil roles (and less frequently, important Town roles) so the player does not draw much attention to themselves and come under scrutiny or even quickly lynched. If you start as Godfather for example, it's a pretty good strategy to try to have a low profile. Refrain yourself of talking too much during the day or start wildly accusing people, since that draws suspicion.
However, you must say something or else you draw suspicion again. You need to search a balance between stating something obvious in the chat and saying nothing.
Always follow the Sheriff's or Investigator's word unless they're accusing a fellow Mafia. If the Sheriff seems determined of the guilt of the fellow Mafia member do not try to minimise the Sheriff's or Investigator's claims because that will once again make you suspicious.
If you're lucky and the Investigator's or Sheriffs don't think your suspicious, they won't be inspecting you till late game. This is very interesting, because the Town will be weeding out any contending Serial Killers or Arsonists.
This strength of this strategy is that you let the Town kill themselves.
When 2 dogs strive for a bone, the third one swipes it away
The tactic is very chance dependant. If you happen to be inspected by a Sheriff the first night, it's pretty much over for you.
Pray that you don't get inspected and try to gently push the Sheriff's into the direction of someone else.