Wittgenstein's philosophy can be divided into three parts: the Problem, the Cause and the Solution.
The problem is a view about what is wrong with much of contemporary philosophy, and which has its roots in the Early Modern period. The problem can be stated simply: It is the attempt to understand human life in terms of the categories of the natural sciences, i.e. the new sciences as discovered in the 16th and 17th centuries.
Here is an example of this problem: identifying mental states with brain states. The appeal of this view is the hope that by explaining mental facts in terms of the purportedly more basic physical facts - that of the brain - the mind can be understood in a objective way. The fundamental thrust of Wittgenstein's view is the idea that this kind of "explaining" is vacuous, as it has only the form of an explanation (with the mental phenomena being re-described in physical terms) but doesn't really provide an explanation. Saying we are conscious because we are in a certain brain state is like saying it rains because the rain-God is making it happen. In the rain case, one is taking a certain form of explanation ("Why is the grass wet?" - "Because John poured water there") which we are familiar with in certain situations, and applying it to a different situation and assuming it must be applicable ("Why is it raining?" - "Because God is pouring water"). Same in the consciousness case. We take a form of explanation from one proven area ("Why don't we float off into the air?" - "Because we are made up of atoms to which gravity applies") and apply it to a different situation ("Why do we have any consciousness?" - "Because our brain is made up of atoms which move in certain ways.").
The cause seeks to explain why the problem is so persistent. Why is it that we seek to understand human life through the categories of the natural sciences? Wittgenstein's answer is: it is because we are mislead by language by the surface similarities of how we describe the phenomenon and the types of explanations we are seeking. In effect, according to Wittgenstein, there is a kind of mistake we keep making because our minds and habits are just set up in the way to keep making such mistakes.
The solution seeks to explain how we can overcome the problem. Since according to Wittgenstein, the cause is our being mislead by language, the solution is not being mislead by language. We have to bring words back to their everyday use. We have to resist the bewitchment of language, etc.
Wittgenstein was profoundly right about the problem. But his view of the cause and the solution are completely vacuous. In fact, Wittgenstein's answers to the cause and to the solution are themselves perfect examples of answers which have the form of an explanation but which don't explain anything. "Why is it so tempting to understand the mind in terms of the brain? - Because it is a temptation intrinsic to our lives and language." "What can we do to avoid this temptation? - We have to avoid the cause which draws us into the temptation." Really, that's the best you got Wittgenstein? Thanks, but no thanks. That's not very helpful.
There are actually much simpler and clearer answers to the cause and the solution. Wittgenstein didn't think of these possibilities because of his generally a-historical and a-institutional approach to philosophy. But if we don't treat the problem as some profound battle of the soul to avoid bewitchment by philosophy, but treat it as just a normal sociological, historical and institutional problem, then very different answers for the cause and the solution present themselves.
A society in order to be held together must have a narrative about how its members are bound together. Here subjectivity brings the threat of anarchy and social dissolution. An objective shared narrative is the glue that holds the people in the community together. Call this social objectivity.
In the middle ages in the West the institutional mechanism for social objectivity was the Church. As in all pre-modern societies, any story about the objective nature of the world had to fit into the dominant narratives of social objectivity. Hence the claim that Copernicus must be wrong because it goes against the Church approved view of the place of humans in the universe. In this kind of criticism, the need for social objectivity is treated as the primary and ultimate basis for any objective claims, as if any claim which threatened to dissolve the existing social objectivity structures is not only a social threat but not objective in any way. Call this the primacy of social objectivity.
It is easy enough to see why societies would be in the grips of the primacy of social objectivity. Imagine a group of hunter-gatherers moving in the desert, and one of them says, "It is morally wrong to kill animals; I am going to be a vegetarian, and so should all of you." This wouldn't happen in reality because the move to vegetarianism totally unravels the social cohesion of the group, since all of their habits and structures are premised around hunting. And if the social cohesion is unraveled, then they are not in a position to actually talk to each other; they can now only turn on each other as enemies. Hence if some member of the group refused to eat meat or kill animals, that person would have been ostracized or killed, because without some basic social cohesion there can be no sustained disagreement in the first place.
No matter how sophisticated or modern a society become, the primacy of social objectivity still holds. In order for society to tolerate a variety of views, it must have in place a narrative of social objectivity which can allow for such variety of views to flourish.
On one story of the change to modernity, in the middle ages everything was about social cohesion and the power of the church, and then with the rise of modern science and democracy, truth and individuality became paramount. In an Enlightened society, the story goes, objectivity comes apart from social cohesion; all that matter is the truth itself, independent of any social constraints. On this story, with the rise of modernity we ditched the primacy of social objectivity.
This story is a myth. Some Europeans four hundred years ago listened to Copernicus and Galileo, etc. not because they broke free of the need for social objectivity, but because the old narrative of social objectivity wasn't working for them, and so they were substituting it with another form of objectivity. Now it was not God talk that was doing the work of social objectivity, but talk of atoms and blood vessels and do so forth. This is the project of the early modern philosophers: to argue against the old model of social objectivity, and to put in its place a new model based on the new sciences. And these philosophers were able to do this not because they were somehow intrinsically more Enlightened beings, but because with the rise of colonialism, European society gained enough stability that it could become critical of some of its claims of social objectivity without itself disintegrating as a society.
I am not saying here that modern science is just another attempt at social cohesion, and has no deeper objectivity. I am making the weaker claim that even given the idea that Newtonian science has nothing intrinsically to say about how society should be organized, proponents of such a science faced the task of giving a new story of social objectivity. This lead to the myriad kind of projects of that Early Modern period, with people trying to find all sorts of ways to make modern science and religion compatible, or not.
As the grip of the old Biblical social objectivity story waned, a new story of social objectivity had to be put in its place. And one main alternative that was put in its place: social objectivity can be read off of scientific objectivity. If not the social structures which can interpret God's word, then what can hold society together? An answer: the very structures which hold the scientific enterprise together. On this view, given that the idea that we are all made up of atoms is the ultimate truth, then any truths of social objectivity must themselves be read off the more basic truths. And this set off the philosophy project since then: Can social objectivity be grounded in natural science, and if so, how? This was a pressing question because, as the power in the society was moving from religion to science, the most pressing social question for science was whether it could provide the requisite social cohesion required for society. Maybe science will help us discover the nature of the cosmos and our bodies, but would it destabilize us as a society in the process that we as a society would fall apart?
The above explains the grip of the idea that "mental states are brain states". This is not so much a scientific discovery, or even a "purely" factual statement, as much as it is a rallying cry for the claim that social objectivity can be grounded in natural science.
If we lived in a society that was primarily structured around God as the locus of social objectivity, then we would say things like "mental states are a part of the soul", where this is basically a way to say that one's growth as a human being (the mental states one has) can only be understood in relation to God, and so in relation to the social structures which are our connection to God.
Wittgenstein's central idea can be put this way (and he wasn't the only who came up with this idea): just as the need to defend God is bad for doing natural science, so too the need to defend natural science is bad for gaining a better understanding of human life. The clergyman who denies that he is only trying to defend God, but claims that it is an open objective issue whether humans were created six thousand years ago is not seeing that he is trying to have a political argument through scientific language, and so in the process is making a hash of both politics and science. Similarly, those who think they are defending the role of science in society by claiming that it is an open issue whether the is the brain are not seeing that they also are trying to have a political argument through scientific language, and in the process they also are making a hash of both politics and science.
Given that human beings are political beings, can the nature of human beings be understood independently of any particular political or social perspective? I think not. This is the question in front of us. What is it to understand human beings objectively? We can't just rely on old narratives of Gods to provide such understanding. Nor can be try to simply shoe-horn our objectivity into the kind of objectivity through which we have understood gravity or even DNA. So what kind of objective understanding can we humans have of our ourselves?
Wittgenstein got as far as clearing the ground so that this question can be seen perspicuously, but he had no positive answer.
The issue is ultimately one of institutional structures. The problem with the middle ages was that every kind of human endeavor had to go through the structures of the Church, and after awhile the Church couldn't bear that weight and so alternate structures of science and democracy had to be created. But now a similar problem arises, where we are trying to fit every kind of human endeavor into the structures of academia, and it can't bear that weight and is crumbling in the process. Just as we needed structures beyond the Church to understand the natural world, we need structures beyond academia to understand the human world. We can't treat understand the mind as itself a purely non-political, non-institutional issue, the way we can with understanding rocks or stars.
This is why anytime a philosopher of mind or psychologist says "we are on the way to getting a basic understanding of the mind, and we are making great progress in understanding the brain" I think to myself, what institutional structures are most benefited by this kind of claim? The more that philosophers denies he is saying anything institutional, but is only making a claim about the objective nature of the mind, I think to myself, this person is so set in the existing institutional structures that those structures are invisible to him, and he is mistaking not seeing the structures with having transcended them.