What the compatibilist is saying is that free will -- real free will
-- is compatible with determinism. The reason the metaphysical
libertarian thinks they're not compatible is that he's adopted a
mistaken analysis of what free will is or would have to be.
Here's a first approximation to the compatibilist analysis: Suppose we're talking about somebody whom we'd describe as free and responsible for her acts. She has certain raw capacities, given by nature, which have been shaped through the course of education and experience into an adult's information-gathering, reason-considering, action-guiding capabilities. For the sake of having a short name, call these capacities, or rather what underlies them, her deliberative machinery.
In the normal case, her actions depend on the output of her deliberative machinery: If she decides to do something, she does it; if she decides against it, she doesn't. She can also (often) defer a decision pending the availability of further information or the outcome of further consideration.
What, then, defeats a claim that she was acting freely or responsibly? The compatibilist says that it is not causation that defeats ascriptions of responsibility, but rather compulsion -- which is a matter of being caused in certain ways. Compulsion is not the same thing as causation; it's a narrower category. She can't jump 17 feet straight up right now, and there's a perfectly adequate causal explanation for that, but that doesn't mean she is compelled not to jump 17 feet straight up. Compulsion is something that interferes with what a person would otherwise be able to do (or perhaps would have reason to do). It makes sense to talk about it against a background of assumptions about the person's uncompelled abilities. It makes no sense when we are talking about the full causal background that includes all of the person's abilities. Basically -- lots of complications apart -- there are two things that can defeat or undermine ascriptions of responsibility: The agent's deliberative machinery may be bypassed or it may be short-circuited.
It is bypassed if it simply has no effect on the behavior. Paradigms of this are cases of sheer physical compulsion. She starts across town to keep a promise but has a flat tire. Or she trips and falls in a way that injures someone. When things like this happen, we say she is not responsible because what happens is a matter of (merely) something happening, not a matter of what she did. Her deliberative machinery doesn't enter into it. There was no decision on her part to break the promise or cause the injury nor any consideration of the merits of injuring or promise-breaking. (In at least some cases, there is room to think that there are psychological analogues -- a person has or suffers from some compulsion that drives behavior independently of conscious intentions or reasoning or deliberation. In many of these, however, matters are mixed: the person with the compulsion may have collaborated in coming to have it or be able to take steps to avoid being in circumstances where it is exhibited.)
The deliberative machinery is short-circuited if it plays a role in controlling behavior but not the normal kind. A paradigm here is response to a threat. She makes a decision and the decision controls her behavior, but the only reason her decision was to turn over all her money to you is that you threatened her with "your money or your life!"
But suppose the deliberative machinery isn't interfered with, isn't bypassed or short-circuited -- which is surely the normal case. Then, she can control her actions in the light of reasons and information she has and can direct her actions to the satisfaction of her desires and the achievement of her goals. If she should wonder whether she has enough information to make reliable judgments about what to do, she can investigate further. If she should come to question her goals or desires, she can reconsider them and, if she finds it appropriate, can alter or change them (or undertake steps to bring about their alteration, such as acquiring new or breaking old habits, undergoing therapy and dozens of other possibilities.)
Her life is under her own control, shaped by her own goals, and even her goals are not simply given or something she is stuck with -- they are what they are because of the thinking and deliberation she has (or hasn't) done about them. In short, she is free, responsible for who and what she is and for the actions that flow from who and what she is.