What and why is "perception of the dimension of nothingness"?

In simplified way. This sutta explains "dwelling in emptiness" which is the same thing as "release" or "liberation" or "nirvana".

It explains it by example, step by step, going from gross liberation to subtle liberation, to absolute liberation. Liberation from what? From mental disturbances.

All liberations except the final are not real liberations. They still have disturbances. But they can be useful to understand the real liberation. In fact all these examples are not supposed to be real exercises. They are only meant to be metaphors to help the Buddha explain and the student to understand the real liberation.

So first Buddha says, imagine that you are in the middle of jungle. What disturbances does this liberate you from? It liberates you from all noise, conflict and issues that come from the contact with people.

But jungle still has its imperfections, like trees and rivers and mountains. So now let's imagine that your are standing in the middle of the flat earth, stretching in all directions. This liberates your mind from disturbances caused by the jungle.

It is important to understand that this is a metaphor. Buddha is not saying we should go to the jungle or to imagine that we are in the jungle or to imagine that we are standing in the middle of the flat earth. Buddha is talking about dukkha and about asavas. He is talking about inner conflict and inner disturbances. Buddha is saying, just like we go from the city to the jungle, to liberate ourselves from the noise, we should first implement the five precepts, to liberate ourselves from the gross disturbances.

As he goes on like this, Buddha step by step comes up with more and more subtle mindstates that have less and less disturbances. Meaning, as we progress in our practice, we get rid of smaller and smaller conflicts, neuroses, intoxications, and obsessions. All dukkha without exception is caused by an inner conflict between "is" and "should". As we go along, we keep noticing and removing even the most refined and subtle attachments, getting rid of even the smallest traces of dukkha, until we enter complete suchness.

Buddha illustrates this step-by-step refinement using the Four Formless Absorptions. Floating in the middle of empty infinite space is more refined than standing in the middle of flat earth without jungle. The image of infinite mind is even more refined than infinite space. The image of nothingness is more refined than that of infinite mind. And the image of neither-perception-nor-perception is even more refined. This does not mean we should actually practice these mindstates! They are here only to illustrate the progression.

The point is, we are looking for a perfect mindstate, and as we're achieving more and more refined mindstates with less disturbances, less conflict, and less dukkha - we realize they still have some dukkha left in them, so we keep going. And so we get to "theme-less concentration". This is known as Diamond Samadhi in Zen, meaning it is the type of clarity or wakefulness that does not depend on any theme or object. Because if we concentrate on any theme, our concentration is limited. Instead, with theme-less concentration we are open to everything that happens, we can look around without losing our clarity of mind, without attachment to any experience. Since we have no attachment, we have no disturbances in our mind. Since we have no disturbances, we have no dukkha.

And then, if we are to remain truthful with ourselves and logically consistent, we must realize that even these theme-less concentration is an artificial mindstate. We are still making effort to maintain it. Since we are still making effort, the state is not permanent. The moment we get distracted or disturbed we can still lose it. Even if we have perfect will power and concentration, the moment we get sick or wounded, we can lose this mindstate. Or we simply lose it when we fall asleep and wake up in a regular state.

Whatever is fabricated & mentally fashioned is inconstant & subject to cessation.

And so we reach the point of final release. We let go of any attempts to achieve the perfect mindstate. The moment we do that, we realize that our very intent to have the perfect mindstate was the last attachment that generated inner conflict that kept us in dukkha. When we let go of this final attachment we enter complete "suchness":

Whatever remains, he discerns as present: 'There is this.' And so this, his entry into emptiness, accords with actuality, is undistorted in meaning, pure — superior & unsurpassed.

This suchness manifests as complete acceptance of "the world" and spontaneity aka self-acceptance. It manifests as perfect unity or perfect love or perfect freedom. There is no sense of "I vs. world" anymore. No sense of self-control. But because of all the previous training, this lack of self-control does not lead to any negative behavior - because we no longer have any attachments that would manifest as Want, Hate or Confusion. So thanks to the previous training, we can be completely free and spontaneous, not relying on any effort, without running into any kinds of issues.

So you train and train and train, removing all external and internal obsessions, rejections, attachment, and sense of inner conflict, removing all dukkha - achieving more and more subtle and conflictless/dukkhaless mindstates - until you just let go of it all, let go of any effort and any notion of realization, let go of all worry -- and that is the final release.