Monster Closets are a way of subtly getting monsters into a level without having them visibly teleport in out of thin air. They are generally used instead of having pre-placed enemies out in the open as a method of surprising or startling the player. They rely on a "trigger", such as an item or plot element, that once interacted with, activates the monsters. They might simply appear in an area the player cannot see, or a previously-locked door might burst open with a powerful monster behind it, or sometimes the term is almost literal, having segments of the wall quietly open, allowing the enemies standing in the hidden room behind it to wander into the level. This concept is like a combination of the monster chest and cheap scare concepts.
This is popularly used in horror games to startle or scare players; by having monsters suddenly appear in an area behind the player, preferably an area the player has already been through, it will be more surprising when monsters have suddenly returned than if monsters had come at them from a new area. This concept has been utilized many times, most infamously in Doom 3, in which any conspicuously-placed item was more than likely the trigger to a monster closet, and players are advised on the higher difficulty levels to skip health and ammo unless they desperately require it, to minimize their encounters with monsters. Notably, Resident Evil contains a literal "monster closet": the player can locate the famous Keeper's Diary in a bedroom; the diary records the thoughts and experiences of a man unknowingly transforming into a zombie. Once they finish reading it, the zombie that man became suddenly bursts out of the closet behind them.
The concept of monster closets has, of late, fallen out of favour with game developers. Now, other tactics to bring frights to the players are more commonly used.