"How can one think what is not the case?" is a typically troubling question in philosophy. We know perfectly well that it is possible to think what is not the case, but we have trouble explaining exactly how is it possible, as if there were some mental mechanical explanation that we haven't quite understood.
Since we can think of things and facts, we mistakenly assume there must be "objects of thought," and wonder how something that does not exist can be such an object. Perhaps the objects of thought are "shadows" of things or facts: mental objects that somehow correspond to things or facts. But how can I recognize a certain shadow to be the shadow of a certain fact? It seems there is some act of interpretation by which I interpret the shadow as being a shadow of a certain fact.
We could imagine a convention according to which we follow the stem of an arrow rather than the point: there must be an act of interpretation involved in reading "—>" as "go right" instead of "go left." In turn, this act of interpretation could be represented as a sign, perhaps as another arrow showing that the previous arrow means "go right." But then this second act of interpretation stands in need of interpretation, raising the question of where the chain of interpretations stops.
Our inclination is to say that what a sign says can be interpreted, but that what the sign means does not require interpretation. Though we may be deceived by grammar into thinking that "meaning something" and "saying something" are analogous, what something means cannot always be represented by signs. When I say, "I'm delighted to see you," whether I mean it or not is determined by my tone and attitude, and not by certain words in my head.
The notion of a "shadow" of a fact comes from the assumption that a fact must be present in our minds if we are to say we are thinking of it. But this assumption leads to the unsolvable difficulty of how the mind is able to interpret this "shadow" as representing a particular fact. The assumption that shadows of facts exist in our minds comes from a particular form of expression. We say things like, "when I said 'Napoleon,' I meant the man who won the battle of Austerlitz." By this we mean that we said a word, and partially defined that word by something unspoken, something "in our heads." There is nothing inherently bad about saying we something "in our heads" as long as we recognize that this expression is metaphorical.
Wittgenstein does not say there are no or processes associated with thought or meaning, he just dismisses the assumption that there must be complicated mental states. No distinctive activity of meaning what we say necessarily underlies all speech. Wittgenstein calls "meaning" an "odd-job" word, one that serves a number of different important purposes. We will have no luck if we look for the one distinctive process of meaning that exists in the peculiar medium of the mind.