First, what is a quantum state? A "quantum state" is all of the information you need to completely determine a system. If you know a quantum state, then you know everything that there is to know about what it describes. And if we take all of the particles that constitute your body to be "the system", then we can teleport it to a new location. Of course, something as large as a person will probably never happen -- right now we're just doing one particle systems (although a wavefunction can describe as many particles as you want).
The news article says that "matter isn't transported, just the state". This is kind of misleading. There's a property in physics called indistinguishability. Electrons are an example of something that has this property. What this means is that you cannot tell two electrons apart. They are identical in every way (coarsely put, electrons are just an excitation of the same "electron field"). This indistinguishablity leads to some awesome physics (see http://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statistic...).
So it's meaningless to say that the original matter has or hasn't transported because there's no such thing as the "original matter". If the state of the "new matter" is the same as the old once was, then for all intents and purposes you have actually teleported that object.
But if you have teleported the state, haven't you just cloned the object? Aha! Nature's too tricky to let you pull that off. There's something called the no-cloning theorem which proves that it is impossible to duplicate a quantum state -- you can only move it around.
And to clarify one other misconception, quantum teleportation does NOT allow faster than light information transfer. You still require a classical information channel to complete the teleportation (so it's not instantaneous teleportation in the sci-fi sense).