Historical anecdotes help us connect with history. We need to know about specific events and people in order to really comprehend the bigger picture. Historical anecdotes make it seem more like a story than a history lesson.
Diamond uses historical anecdotes in this book because they tend to demonstrate the points that he is trying to argue. He uses them, then, as "proof" of the validity of his arguments.
A good example of this is the anecdote that Diamond tells about the Moriori, the Maori, and the rest of the Polynesian peoples. By telling this anecdote, Diamond is demonstrating that a group of people who have similar ethnic and cultural characteristics can evolve different types of societies depending on the geography of the places they settle. This "proves" his argument that geography (rather than "race" or culture) is what determines which societies become "advanced" and which do not.
Diamond, then, uses anecdotes because they tend to support his theory and because historical anecdotes are the only major form of evidence that can be given in support of his theories.