How does it work?
When you hit a certain point in a questline, as usual, you are asked to report the progress to someone far away. At this point your character is flagged to see the new version, so when you next return to the area every model, texture that loads is replaced with a new one. If you are inside a building from where you can't see outside, everything may be replaced around you without you even noticing. There are 2 versions of this technology. The first is a simple one where between phases only NPCs change, the second is when buildings, objects even the sky could change. This is being referred to as a world state change.
What's it used for?
It's an awesome story-telling device, it allows the world to become dynamic, it allows an area to be re-used to function as a new one. Whereas earlier constructing a new building in a town required client patching, phasing allows the same effect to be seen anytime, for anyone. Did you help a faction conquer an area? When you next return they'll have a camp set up with vendors and other services and all the bad guys are gone! The same area now serves a different purpose reflecting your earlier work.
What is phasing?
It's a new technology used throughout Northrend and in Ebon Hold. It allows you to play in a different version of the same area temporarily or permanently, but it differs from e.g. a "shadowy realm" used in many quests, because there you are alone, usually everyone sees his own unique "shadowy realm". With phasing you may share the new version of the area with other players. An early form of it was used on the Isle of Quel'danas to create the phases of the basecamp expanding.
Are there any problems with it?
If you progress in a story to a new phase, you are removed from the previous one making you unable to see or help players in the earlier version. So far most areas that use phasing technology have 2 or 3 phases, and each of them are sufficiently long (or short) to make sure there is enough players in each phase to allow player interaction. Another issue is that creating phases, especially world state changes seems to take up a lot of time, and cause problems like the same mob in a different phase doesn't behave the same way, since it is in fact a different mob.
An example of the first version is when you free Gymer, King of the Storm Giants in Zul'drak, all the regular 8k HP mobs in the lower Zul'drak area disappear and new 50k HP elites (and some bosses) appear that are more suited for Gymer's unique powers.
This allows players to not interfere with other quests with their super-powerful new friend, but if someone else is also controlling Gymer, you can see them and you can help each other. The entire story-telling of the Death Knight starting experience is based on phasing, the second version of it. At first everything is at peace in the Scarlet Enclave, but as you progress through the story, buildings are set on fire, NPCs move around, an army appears or a battle takes place. All without interfering with each other.