After you beat the champion of Kalos when you go home and examine your bed you have the option to sleep in it and when you do you will be in a nightmare that you can't escape from. You will start your journey over again with a twist, you will get the same starter you choose but there are few Pokeballs because stores don't sell them.
This "new game" will follow similar to the main game but with more scary things due to Darkrai's nightmare. Everything will be dim and dark everywhere, similar to how you only have a small field of vision in dark places like the Rock Tunnel, except that you can see everywhere on the screen, it will just be hard to see because it's a nightmare. Wild Pokemon and trainer AIs will be much smarter.
The result of the evil teams is you losing to them, but you still continue to the Pokemon league. Once you defeat the gym leaders (who are evil) and the Elite 4, instead of battling the champion you battle Darkrai. He can't be defeated, only caught. Once you catch him you wake up and your mom and friends are by your bed and are delighted that you are finally awake. In order to get Darkrai you have to go to the Pokemon Lab with a Munna or Musharna to have it extract your dream and then Darkrai will be yours!