The game takes place in 1987, after the events of FNAF2. You play as Mary Schmidt, who was also a worker at the robot factory (from what we can assume) The current lineup of animatronics are Candy, Cindy, Chester, Blank (although he is from the previous generation), and The Penguin. These animatronics are friendly to children, but act creepily to adults, just like the animatronics in FNAF2.
Mary fell asleep on the job at a robot factory, many months or prior to her job at Candy's. This led to several children being murdered in the robot machine RAT (maybe). He was supposed to be stopped by Mary, but she wasn't awake to do that.
The parts that are covered in blood were used to make the "toy" looking animatronics, hence why the Puppet points to them after the murder, in a seemingly "you did this to them" manner.
The children who died in the factory have haunted Mary for "killing them", even though RAT (maybe) was the one who did the actual crime. However, as we can tell from the cutscenes, there is no evidence that RAT did it because the cameras were not recording the instances. Puppet seems to represent Mary and her guilt, due to the "MY" and "YOUR" being used synonymously.
Some time After the incident (presumably months) Mary took the job at Candy's, only to tamper with them the next week and get fired.
On rare occasion when viewing CAM 03 the posters will change to say "YOU FELL ASLEEP" and "IT IS YOUR FAULT", hinting about Mary falling asleep on the job.