Diamond argues that Eurasian countries had much more widespread writing. These countries had literate bureaucracies and generally had a significant portion of the general population that was literate. This meant that writing allowed these societies to advance more than others in terms of political structure, economics, technology, and other things.
This fits well with Diamond's thesis because it shows the importance of having discovered writing early. Societies that did so were able to spread writing among their population and really use it to make themselves more powerful. This allowed them to use writing to help dominate other countries that did not become literate as soon.
Diamond's major thesis is that the earlier adopters of technology (chosen by geographical factors) were the lucky ones. These were the societies that became powerful and dominated the others. Writing was one of these technologies whose early adoption helped make Eurasian societies dominant.