Diamond provides a much more detailed description of the process of determining dates than we can in the limited space here. In Chapter 5, for example, he tells us about the process of carbon dating.
Basically, he says that archaeologists excavate sites where humans are known to have lived. When they find seeds from domesticated plants or bones from domesticated animals (he says domesticated plants and animals tend to be morphologically distinctive, so they can be told apart), they know that domestication had started and must try to date it.
To date these seeds or bones, they use radiocarbon dating. Diamond describes this at length in Chapter 5, page 95 in the paperback version of the book. Basically, we know how quickly carbon 14 decays and becomes carbon 12 once an organism dies. By looking at the ratios of carbon 14 to carbon 12, scientists can know how long the plant or animal has been dead.