Before you try to get here, make sure you have beaten the cupcake minigame first. On Night 5, get to the point where Baby guides you through the Funtime Auditorium. When you get back to the Funtime Auditorium, immediately turn up-right (W+D keys). A voice should have said "Access Granted". You now have entered the Private Room. If you don't complete the cupcake minigame first, the voice will say "Access Denied" and you will get jumpscared by Ennard.

(Thanks to CoolKoopa15 from GameFaqs for this guide)

UPDATE: In the mobile version, when you get back to the Funtime Auditorium, immediatly press and hold the right side of the screen. You have now entered the Private Room.