In my view/understanding, answering your question requires understanding the information loss paradox. This requires that we introduce Hawking radiation, and try to imagine how we would actually go about quantizing fields inside a black hole.
By the no-hair theorem, black holes have only three properties: mass, angular momentum, and electric charge.
The problem is that there is tension between the no-hair theorem and quantum mechanical unitary evolution.
On the one hand, the no-hair theorem says that an outside observer can only really measure the three properties you mentioned of a black hole.
On the other hand, there is still extra information associated with a black hole. It is the information inside the event horizon. For example, if you throw an encyclopedia into a black hole, then the information on the pages of the encyclopedia is inside the event horizon.
That isn't a problem so long as the information stays inside the event horizon, inaccessible to the observer. However, once we take into account Hawking radiation, then black holes will shrink in size, and, it is widely thought, evaporate. (Although one logical possibility is that you're left with a "remnant" that still has all the information inside.)
Now combine the black hole evaporation with quantum mechanical unitary evolution. At least mathematically, it should be possible to take the final state of the system, after the black hole has evaporated, and evolve it backward to infer what happened inside the black hole before it evaporated. In other words, you should be able to reconstruct the words in the encyclopedia (at least in principle -- in practice you would need to measure an enormous amount of subtle information about the Hawking radiation and do a truly mind-bogglingly universe-bendingly epic calculation; the information loss paradox is at its heart an "in principle only" thought experiment).
In other words, there is more information available to the outside observer, than the no-hair theorem suggests, if they are able to wait for the black hole to evaporate.
Furthermore, detailed calculations suggest that the Hawking radiation has an entropy proportional to the area of the black hole.
Where does this information live, while the black hole has not evaporated?
Well, in a normal quantum field theory, you would expect information to be localized in small volumes of space. To take a concrete example, imagine quantizing a free field theory on a cubic lattice with $N$ grid points on each side. Then you are effectively quantizing $N^3$ coupled harmonic oscillators, so the size of the Hilbert space grows with the volume, in an ordinary field theory.
In order that the number of degrees of freedom contained in the black hole, be consistent with the entropy of the Hawking radiation that can be used to reconstruct what is going on inside, it must be that the size of the Hilbert space describing the interior of the black hole grows as the area of the black hole, and not the volume. This is surprising, because it is not how an ordinary QFT behaves. It seems to suggest there must be something fundamentally non-local about quantum gravity.
Now, of course, most of these arguments are quite speculative. There is no widely agreed upon solution of the information loss paradox, and there is no solid proof that the holographic principle is correct. However, this is the kind of logic that people follow, and the end result of this chain of logic is surprising, if it is correct.
