What exactly makes a black hole STAY a black hole? I don't understand how, as a black hole gets smaller and smaller from the excretion of Hawking radiation, it retains its ability to capture photons. I imagine there would be a point in its life cycle where its mass/gravity just isn't enough for it to be able to do so and its body could be revealed, and perhaps gain back some of its previous volume from the lack of gravity being able to hold it together as tightly?
I have no formal education in physics yet.
 A: I take it, that this question is actually about how tiny black holes actually behave. That the aim is to get some kind of intuition about what happens. As such, this answer focuses on describing the last seconds of a black hole. The question gets answered, but the context is also illuminated.

Actually, the black hole does not stay black at all. It evaporates. The more mass it loses to Hawking radiation, the hotter the Hawking radiation becomes, and the faster it evaporates. This is a self-amplifying process that reaches infinite power within a finite time. In the last second of the life of the "black" hole, the Hawking radiation becomes so fierce that it carries away more energy than all the nuclear arsenals on this planet , all taken together, could deliver. Magnitudes of more energy! The radiation will consist of ever harder gamma rays, and at some points electrons/positrons, (anti-)protons, (anti-)neutrons, and other (anti-)particles will start being emitted with ever growing numbers. And all of this radiation is coming out of a tiny sphere comparable to the size of a proton!
As such, "capturing photons" is not really a description of the evaporating black hole anymore. It's a freaking source of photons. Nevertheless, this is all Hawking radiation, and does not tell you any more about the inside of the event horizon than any other Hawking photon does that is emitted in the earlier, cooler stages of evaporation.
And anyways, if something does not interact with a photon, the photon cannot tell you anything about it. So, even though the event horizon becomes too small to interact with incoming radiation appreciatively, it does not reveal anything about its innards. The event horizon still remains the impenetrable shroud that gobbles up anything of sizes comparable to the Schwarzchild radius of the black hole and smaller.
A: After writing this answer, I noticed there are a couple alternative explanations that might be interesting to mention, so I'll add them as well.
Explanation 1
What makes something into a black hole isn't exactly how much mass it has, but also how compactified it is. In principle, any amount of mass can form a black hole, as long as you compactify it enough.
The size needed for some amount of mass to form a black hole is know as the Schwarzschild radius. Roughly speaking, if you pick an amount of mass and manage to compress it below the Schwarzschild radius, you'll have a black hole. It is given by a simple expression. Namely,
$$R_S = \frac{2 G M}{c^2},$$
where $M$ is the mass, $c$ is the speed of light, and $G$ is Newton's gravitational constant (which sort of measures how intense gravity is). For example, for something with the mass of the Earth, the Schwarzschild radius is roughly $0.88$ cm, while for the Sun it is about $2.9$ km (I must admit I didn't double check the computation, I'm trusting Google on these numbers, but they are pretty much what I remember).
Hence, the black hole stays a black hole while it evaporates because it is shrinking while it is losing mass, and always shrinking enough so that it is always at the correct size.
Explanation 2
The second way of thinking is a bit less familiar. 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. As it loses mass, gravity weakens and it gets smaller, but there isn't anything behind the horizon to come out.
Edit: the question "What do we mean when we say that black holes aren't made of anything?" later asked for a more technical discussion of parts of this explanation. I suggest checking it out.
Explanation 3
The third explanation might be a bit simpler than the second. Once something falls into a black hole, that's it. It can't come out. Ever. By the very definition of what a black hole is. Hence, as the hole shrinks, there is no way something could come out of the hole to be its "body". That would violate the very meaning of what is a black hole.

This is a simplified answer. Since OP doesn't have formal education in Physics, I might have overlooked a few details and nuances in here, but I did my best to keep the answer as faithful as possible.
A: As simplified answer to your question is this:

*

*It is possible that they stay black holes because only large black holes can exist.

A longer discussion.

*

*Remember that Physics is an experimental or observational science first of all. Theories that are not supported by experiments or observations may be interesting thought experiments at best.

*We have never seen a "small" black hole and can not really know if they can, will or ever have existed. With small I here mean a black hole that loses more in mass over time than it gains (the very basis of your question).

*What we have found out is that at small scale of stuff, the classic mechanics break down and are better described by other theories, often based on quantuum  effects. The so called Schwartschild radius is built on classical mechanics and probably does not hold up for small things, and hence is probably not relevant for "small" black holes.

*Basically a lot of our theories, makes assumptions and speculates around how small black holes could be created and how they would behave. History has shown that sometimes theories get it right, sometimes wrong. Until proven by experiments the only thing we can do with the theories is to consider them as brain games. You may want to learn the theories, but stay away from believing that they are reality, until experimentally shown to be at least possible.

A: This is not a competing answer but rather a supplementary one.
You say "capture photons" as if gravity is so strong that photons cannot escape, like the pop science idea that "escape velocity is faster than light speed."  To me, this is the wrong picture. General relativity tells us that what we call "gravity" is not a force; rather, the presence of mass and energy changes the rules of geometry in the regions near them in ways that make the trajectories of objects curve. You don't fall towards Earth because of a force pulling. You fall because paths are bent in such a way that they progress towards the Earth.
You can think of it like space and time are a big sheet of graph paper, which in empty space is flat and with lines at 90°, but near a mass like the Earth are curved.  They curve more and more the closer you get to Earth, but before they diverge too much from being straight, you would hit Earth's surface. Now, imagine you crammed all the matter in the Earth into a much smaller space at its center, so the lines could continue curving.  At some point they would curve so much, the geometry would change so much, that outward is no longer a possible direction. You could not move further away from the center by changing direction or adding speed, any more than right now you could travel backwards in time by traveling in a certain direction or getting to a certain speed (not even 88 mph).  "Backwards in time" is simply not an available direction.
A black hole is simply an object whose mass is compressed so much that the curvature can get close enough to the center to reach this critical stage (Schwartzchild radius) where outward is no longer possible. But there is no minimum size. Earth could become a black hole if it were compressed as described to under 1 cm.  The reason in reality we do not observe "small" black holes is the only known process capable of compressing matter to those extremes, is the collapse of stars. And this can only happen with a certain minimum mass.
(It is speculated that cosmic ray particles crashing into the atmosphere at extremely high energies may create black holes with the mass of a few atoms, which would Hawking evaporate in nanoseconds.  But this has not been proven.)
A: Some other supplementary points to amplify other answers.
A black hole isn't really a hole in the sense we use on earth.  Its a name, a label.  So let's not get hung up on the name that we use. So I'll call it a BH here.
General relativity tells us that any matter or energy (same thing really for this) bends spacetime.  Meaning it bends the geometry of space itself (among other things).
If you pack enough "stuff" in any space, it bends space enough that there is a region of space where anything whatsoever cannot avoid moving inwards. Literally all directions, every which way, are so twisted they * all * point inwards.  There isn't any direction which lets you go outward, at all.
What this means is, anything whatsoever within that part of space, can only ever go more "into" the center of that part of space, never ever outward.  We say that "all futures (the future for any object or energy) points inward".
Its in that sense it is like having some kind of "black hole".  Its a part of space where no light can emerge from, and like an infinitely deep hole, whatever falls over the "edge" is lost from view forever. You can't ever get near enough the edge to see through inside that part of space, because light itself also falls inward.
General relativity gets complicated when you ask what counts as "stuff".  Because matter and energy count as the same thing, enough energy in a small enough space, creates this BH effect as well. This is why some answers say that enough gravitational energy alone, can create and sustain a BH, with or without matter.
As a BH incredibly slowly loses energy via Hawkings radiation, the amount of "stuff" within it, shrinks.  If it was spread out equally everywhere it might be a problem.  But it isn't.  Its moved to the center and so its now an infinite or near infinitely dense amount of stuff in an infinitely small part of the hidden off area of space.  The boundary can shrink, and there's still always enough stuff at the center to maintain a BH effect, albeit for a smaller volume of space.
