The answer to this question can be found in Chapter 19 of the book. Specifically, it can be found beginning on p. 393 of the paperback edition. The answer is closely connected to the main theme of the book. The main theme of the book is that geography leads to agriculture and agriculture leads to powerful societies. The factors that allowed the Bantu to expand had to do with their superiority in agriculture.
The first factor that allowed the Bantu to expand was the fact that they had crops that were native to their homeland in West Africa and which could grow in wet areas. On p. 394, Diamond tells us that this meant that
…the Bantu were able to farm in wet areas of East Africa unsuitable for all those previous occupants.
In other words, the original inhabitants did not have crops that were as good as the Bantu’s crops and that allowed the Bantu to expand.
When they expanded into East Africa, the Bantu were exposed to iron metallurgy. They learned how to make iron tools and weapons. According to Diamond, the iron tools and the crops meant that
…the Bantu had finally put together a military-industrial package that was unstoppable in the subequatorial Africa of the time.
So, the Bantu were able to expand their influence because they had better crops than the people in East Africa and because they learned to use iron and thereby were able to defeat peoples to the south.