In the Harbringers - Illidan cinematic, the difference between DHs and demons is being discussed. It is said there that while demons will stop at nothing to destroy Azeroth, DHs will sacrifice everything to save it. Not just from the Legion as it would be stupid. I mean, why save the world from one threat only to let it be destroyed by another?
My statement was mostly based on the "The Power to Destroy" quest that Aucald mentioned, but he got things mixed up a bit. The shadowy tormentors were not being sent by the Lich King, he actually kept them at bay. Now though since DKs are no longer under his protection, these shadows are free to torment them. Think of how powerful the shadows must be, being able to reach DKs in the realm of the living, and what they can potentially do to defenseless DK souls entering their own domain. We get a glimpse of it in the Sylvanas short story. Arthas himself was reduced to a terrified child, and regular DKs won't have it any better. As for "why", I'm not sure I remember it correctly but it's likely because of all the atrocities they committed.
Others have already covered this, but take a look at this section of the Wowpedia article on Death Knights: https://wow.gamepedia.com/Death_knig...th_as_a_knight. It has sources.