In superluminal phase velocities, what is it that is traveling faster than light?

Shine a flashlight on a wall. Rotate the flashlight so the illuminated spot moves.

Q: How fast does the spot move?
A: It depends how far away the wall is.

Q: How fast can the spot possibly move?
A: There is no limit. Put the wall far enough away, and the spot can move with any speed.

Q: What is moving across the wall?
A: Nothing. The light that makes up the spot at one instant is unrelated to the light that makes up the spot an instant later.

This is how a wave can be apparently superluminal: we interpret a series of unrelated events as a continuous 'wave'. Group velocity can also be superluminal; even though the individual chunks of energy are going at roughly $c$, the region where they superpose constructively (the 'crest of the wave') goes faster than $c$.

Let me make an analogy which may be helpful.

The next time you're at the beach, watch the waves coming in. You will notice that the point where the wave breaks will often move faster than the waves do. This happens when the waves come in close to perpendicular to the beach.

The effect is that if you know the wave is breaking at point A, you can predict that it will break at point B soon afterward. Note that the short vector gives the wave movement while the long arrow shows where on the beach the wave is breaking. But this is not information that is moving; the wave has the information built into its wave structure:

The same thing applies to de Broglie (or quantum matter) waves which are also superluminal. The short answer is that there isn't anything that is traveling faster than light. It's just an effect of little physical meaning.

Look at a particular local maximum on the electromagnetic field. A small moment of time $dt$ later, there will be a local maximum a small distance $dx$ away. Superluminal phase velocity means $dx/dt > c$. The "thing" moving faster than light is not a thing at all, but a point at which the electromagnetic field is a maximum (or has some other, constant phase, i.e. a minimum, inflection point, etc.)