The answer to this can be found in Chapter 16. The discussion of this topic starts on p. 324 in the paperback edition of the book.
In general, linguistic evidence shows us that the Sino-Tibetan people came from the northern part of what is now China and displaced other peoples in the South. This can be deduced, Diamond says, from the fact that other language families are spoken in widely dispersed areas. Diamond says that this implies that they were once spoken over a more contiguous area, but that other people came and took away some parts of the lands inhabited by the speakers of the other languages. Since Sino-Tibetan languages are spoken across all of the region, Diamond reasons that they were the people who came and took land away from the others.
Thus, we can look at the relative distributions of Sino-Tibetan speakers as opposed to for example, speakers of Miao-Yao, and say that the Sino-Tibetans spread across China, displacing the speakers of other languages.