From the Super Mario Wiki
The Inner Courtyard is an area in Treacherous Mansion in Luigi's Mansion: Dark Moon. Located in the center of the area and accessible from both Grand West and Grand East, this area has a staircase allowing access to the second floor of the mansion.
The door on the balcony to the north leads to the Terrace. The door on the west balcony leads to the West Corridor and the one on the east balcony leads to the East Corridor. On ground level, the door to the west leads to Grand West and the one to the east leads to Grand East.
The area has numerous plants and trees around the outside, balconies to the north, east and west and either a staircase or fountain in the middle.
During Double Trouble, Luigi has to bring two Toad Assistants to the Inner Courtyard so they can be sent back to the Bunker via the Pixelator on the north balcony. However, the stairs seem to have vanished and been replaced by a fountain. This fountain has been made invisible by a Boo, so Luigi has to defeat the Boo, make it visible with the Dark Light and use the Strobulb to make the stairs appear again and let him and the Toads reach the Pixelator.
In A Train to Catch, Luigi has to make his way back here and use the special mechanism to spin the stairs around.