How can brain know about consciousness if it can't be observed outside ourselves?
The fact that we can talk about our own subjective experience as subjective experience, and about our qualia as qualia, seems to prove that it is not true that these things cannot be observed objectively. However, it is also probably mistaken to frame the problem as if it was an empirical question.
The problem of consciousness, the hard problem of consciousness, is fundamentally that there is nothing in the materialist model we have of the physical world which we could use to represent the quality of our subjective experience, i.e., our qualia. If we want to say that the physical world is made of matter, and that the properties of matter include mass, electric charge, magnetic field, etc., then none of these things seem appropriate to explain the quality of our subjective experience, i.e., our qualia.
We know ourselves, meaning we know our own consciousness, which only means that our own consciousness knows itself. The material world that we know, however, is not the actual material world but a model logically derived by our brain from the perception data located in our brain. However, the symbols used to represent this model are nothing but our qualia themselves. Thus, I suspect that the real problem is essentially the old problem of self-reference, which logically obviates the possibility of a rational articulation of qualia in terms of the material world.
Our rational explanations about the material world do not change this situation.
We know our qualia exists because we know them. We certainly believe that the material world exists, but it is equally clear that it is not really as we think of it. The only criterion deciding that the model of the material world produced by our brain is valid is that it should be what allows us to survive in the material world as it really is. A model does not need to be realistic to be effective and useful. We only need it to be useful, not realistic. Thus, the question is not to explain our qualia in terms of our model of the material world, we are probably unable to do that. Rather, it would be to produce a model of the material world in its own terms, not in terms of our qualia, and this is also something we cannot do because all we have is our qualia.
This, however, is no problem. All we really need is to survive in the material world and it seems we do. To achieve survival, we don't seem to need to understand our qualia in material terms. All we need is to understand the material world in our own terms, which are our qualia. Seems to work.
The problem of consciousness is really the problem of the material world, namely, the fact that we cannot represent the material world in its own terms, i.e., in actual material terms. It would not be possible to represent atoms using actual atoms, for example, because we would have to represent an elephant as an actual elephant and it wouldn't fit into our brain. So, our brain can only use symbols and its own symbols, namely, at it is, our qualia. Our brain is the product of natural selection, so its function is probably limited to help us survive in the material environment, not to help us solve metaphysical questions that have probably no effect on our survival.