The antecedent tells us that, according to PW, (□φ)*. Why should that enable us to conclude that φ is true? After all, PW is acknowledged to be a piece of fiction... how can it be legitimate to infer the truth of φ from what this wildly implausible story says about (□φ)*?
But back on p.170 he quoted Rosen's exposition of modal fictionalism, explaining that PW includes an encyclopedia consisting in a list of the "non-modal truths about the intrinsic character of the universe". That is, PW includes all non-modal truths about our world, in addition to a whole bunch of recombinant falsehoods. On the Lewisian story that PW tells, □φ is true iff φ is true in all possible worlds. So the italicized consequent is what (□φ)* means. Now, note that one of the worlds in the fiction of PW is an accurate representation of the actual world. So if (□φ)* is true according to PW, then this entails that φ is actually true, as required.
Am I missing something here?