The Story of Five Nights at Candy's 3 is as follows:
Mary Schmidt, a young girl, is seeing a therapist because she is having nightmares about monstrous versions of RAT and CAT at night. This Fear has taken over her creativity as well, causing her to draw those monsters and little to nothing else. The Therapist concludes that the only way to have Mary get over her nightmares and be a child with a normal life again is to directly conquer the monsters.
Mary enters the Dreamscape, where she is assisted by the Origami Cat, which she and her mother made many years ago. It is also "the last memory of her", revealing that her mother died. While the Origami cat is helpful, knocking him off the desc every night will cause him to transform into a monster and haunt Mary forever as revenge, a fake ending ensuing.
If The Player progresses the story normally, though, the story behind Mary's trauma slowly starts unfolding. When Mary Schmidt was just a child, her father often took her to the Rat and Cat theater because she loved watching them. Every time, she would play hide-and-seek with the Rat or the Cat. One day, at a big show, she played the game again with a group of kids and had no hiding place to go. The Cat had accidentally left the door to the employee's room open, so she decided to go in there and hide in the closet, thinking it was all in good fun. Rat bursts into the room, proclaiming he saw her, only for the The Puppeteer to walk in as well. The Puppeteer notices that the Rat's actor has been drinking too much and says that he shouldn't be drunk on the job, especially around kids.
The Rat's Actor retorts, saying the kids love him. When The Puppeteer repeats his statement, the Rat taunts him in a drunken state, causing the two to fight. In the struggle, Rat's actor is pushed backwards, hitting his head against the edge of a metal table hard enough to crack his skull and kill him. The Cat, walking in again, wants to call the police and tell them the truth, but Puppeteer, not wanting to get arrested, strangles the actor to death in his costume and leaves, telling the police a false statement. Mary blames herself for the Rat and the Cat getting murdered, causing her to see them as horrible monsters. The Therapist tells her to not blame herself for this after she completes five nights, Mary telling the truth about who was behind the murder. The police arrest him, and one final night, Mary confronts him in the form of Monster Vinnie where she completely conquers her fears, leading to one final therapy session deemed successful.