A Mystical Dampener is a spell placed on a location to dampen any magical effects and prevent spells from working there.
In 2017, Arthur Ketch imposed a lockdown on the Men of Letters bunker that trapped Sam and Dean Winchester along with Lady Toni Bevell within without power or water and with their air running out. As part of this lockdown, Arthur added a mystical dampener to prevent any magical attempts at escape, anticipating that the Winchesters and Toni would attempt a magical solution. When the Winchesters and Toni performed the Abrogation Ritual to end the lockdown, the mystical dampener caused the ritual to fail at the last moment, restoring the lockdown. Toni then realized what Arthur had done and told the Winchesters that magic wouldn't work as a result.
While the procedure is unknown, it is implemented by the British Men of Letters on a location they wish to prevent magic being used in.