From the Super Mario Wiki
The Train Exhibit is a room in Treacherous Mansion in Luigi's Mansion: Dark Moon. Located to the north east of the mansion, it is accessible via the Study to the south. Inside the room, there is a display near the center where the Big Boo boss battle takes place, a train track around the outside with a rideable train and some decorations to the sides that can be interacted with via the Strobulb.
In A Train to Catch, reaching this room is the entire aim of the mission. It has a couple of ghosts Luigi will need to defeat to light up the area, and then looking into the display itself will end the mission. The player can also get the mansion's 13th gem here. After clearing out the ghosts, the player must shine the Dark Light on the missing piece of track in front of the display case. Then, they should get on the train to go around back and collect the gem.
In Ambush Maneuver, the battle with the Big Boo takes place in the display located in this room. Because Luigi has been miniaturised and teleported into the display via the Pixelator, there is no way to exit the display/boss arena or explore the rest of this exhibit in this mission.
After these two missions, the Train Exhibit is irrelevant for the rest of the game.