This is an excellent question. Some good advice on this can be found in the writing of Bill Thurston, some of which I have posted in an answer to this question on Math Overflow.
The opening of the quotation I posted there is particularly telling: "Mathematics is a paradoxical, elusive subject, with the habit of appearing clear and straightforward, then zooming away and leaving us stranded in a blank haze."
This feeling of being abandoned by intuition, or feeling that mathematics is a wall keeping you outside, is common even to older mathematicians. I, personally, struggle with this all the time. When in the throes of a problem about which I have developed an intuitive way to think, life is never better. Other times, though, I feel somewhat lost. The literature looks like a lot of noise that does not promise the sense of meaning one gets from having that intuitive idea that is making sharp predictions that prove theorems.
The point, though, is to keep trying to make your own intuitive way into the subject via SOME channel. Just keep trying. I think this is what mathematics is, at its heart: trying to develop these human-friendly ways to think about mathematical problems. This trying is connected to a certain feeling of freedom to play with one's surroundings using one's natural capacities. This essay may help with some first steps to try.
Extension of Answer:
There are other things to read that may be helpful. One that comes to mind is Hadamard's Psychology of Invention in the Mathematical Field.. In this essay, descriptions of several mathematicians' thought processes are described. Again, read as much Thurston as you can, as well as some essays by Gowers and Tao's blog, etc.
As a starting assumption, we can take that mathematics resides in the minds of living mathematicians. The lifeblood of mathematics is how mathematicians think about the problems and propositions that are recorded in proofs and calculations. To differing degrees, this can appear as residing inside these proofs and calculations. (Why else would someone as great as Hilbert even dream of suggesting his formalist programme?) Mathematicians have the capability of making associations and analogies that lead to new mathematical connections. Most such ideas occur in the minds of mathematicians, minds which have evolved to have various modules for interacting with the "external" world via sensory data. The spatial sense, perhaps with the sense of touch yields geometric intuition, the sense of time, language and hearing (think music) gives rise to an algebraic intuition. The beauty of these is that they are not evenly subdivided: an algorist's sense of formal beauty sometimes can come from a "geometric" appreciation of formulas, and there is certainly a temporal aspect of drawing a diagram that might communicate more than seeing a static picture.
Some mathematicians have cast mathematics as a dance between the continuous and discrete, or perhaps "potential" versus "actual" infinities. This is certainly related to the interplay of this sense of time and space.
In Hadamard's essay above, he reports that most mathematicians don't think verbally, but instead somehow fashion "vague images" to manipulate in the mind. One mathematician desribed these as "muscular contractions". Interestingly, George Polya (who wrote 'How to Solve It') was the only mathematician surveyed by Hadamard who reported working largely verbally. Others reported working with some sort of semi-linguistic thinking.
Most mathematicians would advocate for computing "to get feelings" for some bit of mathematics. Why does this work? Why does computing give us feelings? (See Whitney's excellent essay linked to above for a bit on this.) Some mathematicians can't think without a pencil and paper nearby on which to scribble. Other mathematicians have to go for a long walk to try to imagine what a computation is likely to "look like" before sitting down to actually compute (otherwise they would immediately feel overwhelmed by the complexity). I think it was Gowers (please, someone, correct my if I'm wrong) who wrote that a mathematician tries to "imagine the outline of a proof, no matter how vague".
We all want a sense of "gestalt" to the mathematics we do. We want to feel the entirety of the proof we are working on as a whole in our mind. That said, we have to work on the pieces, to compute and to piece together a chain of steps. (This feels somewhat, again, like the reconciliation of "space" and "time".)
My personal position is that to feel like we are "living in the mathematics" and not "held outside the walls", mathematics must be done with the whole person (or at least the whole brain), in the sense that the fullness of experience we undergo while struggling with a problem makes impressions on our mind that give the problem a "face". We can move around in our minds to see the different parts of this face and see how calculations are to work out before we even begin them. We build this "face" by trying to do mathematics in deeply engaging ways over a very long period of time. We know the face of our most beloved problems like we know the face of a loved one. We know its every contour and yet still long to know more. Each "feeling" we gain from a calculation or more bizarre attempt to understand (think of Terry Tao rolling around on the floor of his office to understand a PDE…he reported doing this somewhere on MO) is stored to be accessed later. All the people we talk to, pictures we draw, things we smell while working, places we worked, pain we endured while working etc… all of these things are rolled up and embedded in OUR mathematics. To a mathematician, genuine mathematics is an expression of the whole self and cannot be anything less.
How to develop human-friendly ways of thinking about things? Struggle mightily to be able to explain and use them in as many ways as you can. As you attempt to understand a proof, the total of the attempts you make to get used to the reasoning, the myriad angles you try in order to "understand" it, all the angles you try to see it from, all the failed attempts, will give it a face that will become part of you. I don't think this process ever ends. You just have to fight to keep trying. I think there is a lot to say about Whitney's "freedom" in his essay, in this regard. As the cares of life multiply, the ability to fight for mathematics is threatened. If you can keep your life simple enough to protect this freedom, this is probably the best way protect the ability to continue.
I apologize if this is all too "subjective and argumentative"
In light of the bounty comments, I can add a bit more to this.
We may interpret internalization as a feeling that the mathematics is part of us, and not something we are superficially tinkering with external to ourselves without any feeling of understanding. I will share how I activate each of Tao's modes in my own work, phrasing them as specific things to try:
Formalism: Given a problem you are interested in, take the simplest nontrivial example you can think of, and just compute with it on a sheet of scrap paper. Writing down these details formally will give you a sense of the problem that can be felt as a sort of muscle-memory. To amplify this effect, write details on a large blackboard. This will amplify your spatial sense to aid with memory of the written details. Also, speak the formal logical arguments out at different scales, to activate your auditory sense. I find it good to speak out loud details while you are taking a long drive or walk. Try to picture the vague shape of the calculations and rapidly predict how they should look when complete. Roughly, push symbols around on paper and in your mind. Focus on how the symbols interact and the balance of the symbols and their analogy with other computations you have seen before in the literature. Choose different notations. Just calculate. Make formal substitutions. Doing this will contribute to your "feelings" for your problem. Insight here will rely on "formal analogy", in that formulas and calculations that remind you of other formulas and calculations will lead to insights (Read the Poincare essay. His remarks about "Theta-Fuschian functions" discuss such a formal analogy.)
Geometric intuition: Simply put, draw pictures. Try to get away from the symbolic and instead focus on the spatial. Scribble pictures that express the relative relationships of things on paper. If you are working in higher dimensions, try to capture some aspect of the problem in a 2 or 3-dimensional picture. (E.g. if you are working on a closed convex cone in an infinite-dimensional Hilbert space, just draw a picture of a cone and try to think about this low-dimensional analogy.) Gesticulate with your hands while trying to visualize the relations in three dimensions. Try to animate these pictures. Make sound effects (don't laugh, this can help sometimes) for how certain parts move. Even when doing analysis, draw sequences and limit points etc…and not just inequalities. Try to engage your visual sense. This often requires shutting off your verbal sense.
Physical intuition: This is a lot like geometric intuition, but incorporates our sense of force and motion more actively. Try to interpret the problem in a way where physical sense, or the ability to predict what should happen as a result of a physical experiment, can suggest a solution.
Other heuristics: A heuristic is a rule of thumb. Analogy is a powerful tool for mathematics, and some thinkers believe that analogy is the basis for nearly all thought. Mathematical analogy is a comparison of two mathematical structures such that the component parts making up the structures are compared, as well as how they fit together. Any way we can bring abstract mathematics into analogy with sensory experience in such a way that we can more intuitively think about the mathematics can be useful. For example, when we learn to count as children, we are not taught about equivalence classes or von Neumann ordinals but instead witness a manifestation of the abstract in counting stones with our hands. The feeling of understanding that comes from feeling and seeing and hearing and smelling and speaking the numbers yields a sense of understanding. Seeking other analogies to make abstract things concrete is one of the principle tasks of practicing mathematicians. Connecting back to more concrete intuitions is key. This is the role of working with the most concrete examples possible. This said, the ability to rapidly read papers in an area is tied to the internalization of certain key fruitful analogies developed by the leading experts in the field. Talking with these people, you can quickly gain their intuitions and use these intuitions to help make sense of the literature. For example, Sorin Popa once described his "deformation/rigidity strategy" as follows: Imagine a bucket of dark water in which you know there is a bowling ball. If you put your hands in the water and swish them around but never feel the bowling ball, you know the bowling ball has to be where your hands never went. This simple picture (of very crude physical intuition type) maps pretty faithfully to one of the most successful and productive ideas in my subject's history. Of course, a lot of hard work goes into applying the idea. Without the idea, though, the literature can look very opaque to an outsider. Often you can get a lot of mileage from getting good organizing analogies from "the big guys". I've asked an MO question that should try to gather some more such analogies from different areas of expertise in the hope that these will help you.
Language and writing: In addition to the above, I think that writing mathematics down (and writing it up) can help make it intuitive, as it improves memory and harnesses our considerably developed linguistic sense. A turn of phrase or slogan or other written "intuition pump" can help us feel more connected with our problem.
To test whether we are internalizing mathematics is somewhat impossible, as it is a matter of our subjective experience of the world. You know that you are internalizing mathematics when you feel it. It is for this reason that I say it is intensely personal. Your experience and mine are very different, and this is the great benefit of having many mathematicians at work. Our individual experiences provide uncountably many perspectives that may lead to new interpretations and discoveries.
EDIT: Amir Asghari reminded me of this fabulous MO question, which is probably a better answer to the present question than all of what I've written above!