Each NPC Trainer resident has a set number of points, which will change depending on certain functions that can be performed by the player. If an NPC's points value reaches 0, then they will leave Black City.
When using the Entralink feature to visit another player's White Forest, the player can invite people that have left to come to Black City to increase occupancy; this can enable new items to be purchased and new varying Trainers to battle.
"I'm bored of Black City!" means "I want to go to White Forest!". You need to go White Forest and get trainers that want to go to Balck city.