The initial mistake lies in the assumption that when we talk about sensations, we are talking about states or processes. Wittgenstein's denial of private language seems to be a denial that these states or processes exist. He is not denying that there is more to pain than just pain-behavior, but he is denying that we can talk about what lies behind the behavior in any coherent way.
Wittgenstein's analysis of private language is one of the most discussed passages in the Philosophical Investigations largely because there is little agreement even as to what he is trying to say, let alone what conclusions he reaches.
One approach to this section is to read Wittgenstein as responding to skepticism about other minds. The skeptic believes that I can know my own pain in a way that I can never know other people's pain: I base my judgment that someone else is in pain on that person's outward behavior and not on the pain itself. I can never know with certainty that the other person is not pretending to be in pain; further, I can never know that what other people call "pain" feels like what I call "pain." Further, I can never know that other people even exist: perhaps everyone but me is just an automaton built to act human, and lacks all the inner experiences that I associate with words like "pain."
Wittgenstein suggests that these skeptical arguments, and the different responses and refutations to them, are not so much false as incoherent. They are parasitic on the way we normally talk about things like knowledge, doubt, and justification, but do not use these terms in their proper contexts. For instance, in section 246, Wittgenstein says, "It can't be said of me at all (except perhaps as a joke) that I know I am in pain." Ordinary talk about knowledge involves questions of investigation, verification, justification, and so on. I am at a loss how to answer someone who asks, "how do you know you are in pain?" because there is no evidence or justification I can appeal to beyond the simple fact that I am in pain. My own pain is not the kind of thing I can talk about in terms of knowing.
This line of reasoning goes directly against the skeptic, who wants to set up his knowledge of his own pain as a paradigmatic case of certainty, against which my knowledge of other people's pain seems sorely lacking. A skeptic would say that his experience of his own pain is not a clearer, more certain version of his experience of someone else's pain: it is a different sort of thing altogether.
Part of the problem with the idea of a private language is that it tries to set up a way of talking about inner sensations in the same ways that I talk about outer facts. But talk about outer facts is indeed tied up with questions of investigation, verification, justification, and so on: it is part of our ordinary language-games that we can ask questions like, "how do you know?" "on what grounds do you say this?" and so on. These questions do not make sense in reference to my own sensations, so a mark like "S" cannot be taken to state or claim anything in the way that we might want it to. Insofar as we can ask questions, justify, or provide evidence regarding pain, we can only talk about other people's pain and the behavior they exhibit.