What do we mean when we say that black holes aren't made of anything? I have read this question:

It turns out that black holes aren't really objects, but rather regions in spacetime. In fact, this is so true that black holes are what we call vacuum solutions: there isn't matter anywhere in the spacetime. All of the mass of the black hole is there due to effects of gravity itself. Another way of thinking it is that a black hole is so collapsed that its mass is entirely due to gravitational energy.
It is a bit harder to grasp this concept, but once you get it, the rest is simpler. The black hole stays there because it isn't "made" of anything. There isn't a star just below the event horizon waiting to come out. There is nothing there, but gravity.

What exactly makes a black hole STAY a black hole?
While it is widely accepted that certain black hole solutions are vacuum solutions, known black holes are hypothesized to be collapsed (neutron) stars, so the collapsed remnant must be there somewhere (or maybe it transforms into the energy of the gravitational field).
Question:

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*What do we mean when we say that black holes aren't made of anything?

 A: That's one of the things I was thinking about when I wrote the disclaimer at the bottom hahaha. I'm glad you asked, so I can comment on it!
As seem from the outside, the collapse of a star never really finishes. The object just gets extremely redshifted to the point it is no longer visible. One can then argue whether that can or not be called a black hole as seem from the outside. What is sure is: from the outside, you won't ever see the matter stop falling, and hence there surely is still matter there at least on the outside. I'm afraid I don't know the details of star collapse well enough to give a much better discussion. In that answer, I overlooked these details because OP didn't have a Physics background. Considering these nuances of stellar collapse makes the overall picture way more complicated and I considered it wouldn't be suitable for that particular answer.
Furthermore, as other people mentioned in the comments and in other answers, the Kerr and Schwarzschild solutions are vacuum solutions which approximate the exterior part of a black hole extremely well. We can actually get a grasp of black hole evaporation with a Schwarzschild black hole that was not formed out of stellar collapse (Kerr black holes are more complicated), and in this sense black holes are made of gravity alone.
I should also comment on

maybe it transforms into the energy of the gravitational field

This is a more technical way of understanding what I meant by "There is nothing there, but gravity". You have gravity being sourced by gravity itself. However, there is a nuance: gravitational energy is not well-defined in General Relativity. One can't define a covariant stress-energy-tensor for gravity because it is always possible to choose coordinates in which the metric is Minkowski and the Christoffel symbols vanish, and hence the stress tensor would need to vanish as well. Some sense can be obtained by using pseudotensors. Of course, all of this is just a way of interpreting the fact that the Einstein equations are nonlinear.
You might also be interested in this answer of mine: Addition of gravitational fields in general relativity.
A: A black hole twists space and time around so that time (the future) points inwards towards the singularity. All the possible worldlines of infalling particles go in to the central point and then have nowhere else to go. We don't know what happens next - Einstein's theory doesn't say.
Some think they stop there, forming an object of infinite density. Some think an unknown theory of quantum gravity takes over and some new unknown-to-current-physics effect stops the collapse. The singularity is replaced by some string-theory 'fuzzball' object, or something similar. Some think that they 'fall off the edge of space' and disappear. (Like, if you hold a length of rope at one end and wiggle it, you can get a wave to pass down it to the other end. What 'happens to' the wave when it gets to the end of the rope? Where does it go?) Some exclude the singularity not because they think it necessarily actually disappears but just to keep the maths simple - General Relativity assumes for a lot of its working that spacetime is a 'manifold', which cannot have edges or corners. Some think that they squeeze through the point and emerge into some other domain. (e.g. 'Journey Beyond the Schwarzschild Black Hole Singularity' Arraya et al.) Common sense beliefs derived from our ordinary experience of conservation of matter don't necessarily extend to such extreme circumstances. We don't know.
Nor does it really matter, for the purposes of understanding the 'source' of the black hole's gravity, because the singularity is in the future of every particle that has yet to hit it, and causality requires that the future has no effect on the present. The gravity you experience outside a black hole is entirely due to matter falling into the black hole in the distant past, before it reaches the event horizon.
We can draw a picture of the spacetime around a black hole stretched and twisted so that at every point the lightcones point up the page, like in flat Minkowski space. These are Kruskal-Szekeres coordinates. In this diagram, light always travels at $45^\circ$ angles to the vertical, and local time around any point flows up the page.

(To connect this picture to the point of view of a distant observer seeing 'a spherical black hole floating in space', tilt your head $45^\circ$ to the right. The singularity (thick black line) is the timeline of the point at the centre of the black hole, at radius zero, and the event horizon (dashed lined) is at a fixed radius away from it, which diagrammatic scale distortions have expanded into a 'trumpet' shape centred on the horizon.)
You can see the collapsing star that forms the black hole shaded orange on the left of the diagram. Light from it travels up the page at $45^\circ$ to the vertical. Because of the twisting of spacetime, it moves parallel to the event horizon, stuck on the boundary. It is like water is flowing inwards down a drain, and ripples spread outwards at exactly the speed the water is flowing inwards, hanging on the boundary forever.
Observers hovering outside the black hole, or falling into it, can only see events happening on their past light cone - the two lines pointing downwards at $45^\circ$ to the vertical. When you look towards a black hole, you can see its entire history right back to the beginning, apparently frozen in time - it only looks black because the last few seconds of light emitted are spread out thinly over millions of years.
Likewise, they can only see the same region gravitationally. And like the light, the gravity of the collapsing star is also trapped on/near the event horizon. Even though the matter has long since fallen into the hole and met its fate, the gravitational information 'emitted' from its descent into the hole is still in the process of escaping.
The Schwarzchild black hole is a 'vacuum solution', in that it is the shape of empty space surrounding an infinitely dense point mass. It approximates a real black hole, which is the shape of the empty space surrounding a collapsing star. The gravity of the collapsing matter remains behind, trapped forever on the event horizon by the matter's past distortion of spacetime, even long after the star itself has collapsed and fallen into the singularity.
A: The mass is still there, inside the event horizon. The thing that you observe, the black hole, is a region of space which is so curved not even light can escape. But that is why your quote says a black hole isn't an object: the actual region that is black isn't made of anything.
So a small note on it being a vacuum solution. Using general relativity we can write down an exact solution of gravity for a perfectly spherically symmetric body (with the rest of the universe a vacuum). With spherically symmetric I mean that the density of the body only depends on the radius from the center. Because the equations in general relativity are so complicated this is one of the few exact solutions that are known. And actually, it is not a full solution. If $R$ is the radius of the body then we can divide the space into two regions: $r<R$ being inside the body and $r>R$ being the vacuum. For $r>R$ the solution is known at is called the Schwarzschild metric. For $r<R$ the solution is generally complicated and I don't know if an exact solution is generally possible.
For the Schwarzschild metric it doesn't matter how the matter is distributed, as long as it is spherically symmetric. So we could squeeze the matter matter together and the gravity outside would still look the same. That is basically what a black hole is. A bunch of matter which gets squeezed together until the escape velocity at the surface exceeds the speed of light, which results in the matter being squeezed together indefinitely. But from the outside we don't notice this squeezing anymore because the gravitational field doesn't change outside the black hole. In fact, it is not known what happens exactly to the matter inside a black hole. From simply following GR one would expect the matter to get denser indefinitely until a singularity is reached. I don't know what the current consensus about this is. If you know let me know in the comments.
A: 
What do we mean when we say that black holes aren't made of anything?

I risk the answer.
A black hole in general relativity is a spacetime that have gravity (converging geodesics) but no matter (zero stress-energy tensor). The simplest black hole describes the Schwarzschild vacuum metric that represents static spherically symmetric spacetime free of matter.
Now, the point is that spherical symmetry in general relativity is a feature of spacetime and not matter. The matter merely inherits it through Einstein Field equations (EFE).
Imagine a flat Lorentzian spacetime. It has no gravity because its geodesics do not converge. To cause them converging one needs a piece of matter that when compressed to some critical compactness ratio ($r_{S}/R$) gives rise to the initial event horizon at the center. Further compression shifts the event horizon to the surface of the matter ball what is a very sketchy description of the black formation process. The end result of that process is a spacetime without matter disconnected physically by the event horizon from the hypothetical "interior" transient spacetime.
A: It is confusing because he is still using the word"gravity". One must stop thinking of gravity as a force at this point, and remember that gravity is an artifact of time- space. so as time stops, "gravity-space" becomes infinite, and where space stops,  "gravity- time" becomes infinite. But all of this is only describing what is observed with light!
One must, at this point understand that the universe exists beyond our ability to see it with light, As the merging black holes prove with their measurable gravity waves. We measure gravity waves with changes in space/ time,  not with light! So we"see"the wave as it is across time and space,  not light waves. Like the difference between amplitude modulation and frequency modulation. In this case F.M. is the detection of gravity waves using wave interference across space, and A.M. is the detecting light from the star across time.
They both tell the story, but from different perspectives. When you combine the two, you get an extra dimension of data that correlates between them. These data DO NOT EXIST ALONG THE SAME DIMENSION! One must stop assigning all data to the same dimension!
Light is not the end- all,  be- all of the universe. The universe exists beyond your ability to see it with light. As we have clearly defined,  light has limits, remember that when contemplating the universe. Close your eyes, and see the universe in your mind,  without the light that enters your eyes. The limits of light are the deceptions of the nature of the universe. You must "see" past them.
