Yes, the causal link that Diamond establishes between domestic plants and animals and literacy and steel production does at first seem rather strained until we look at the chain of reasoning behind it. In this brilliant book, Diamond argues that Eurasia became dominant in the world because it had the largest number of domestic plants and animals. This, in turn, allowed Eurasians to give up their hunter-gatherer lifestyle and become sedentary, growing through establishing food surpluses thanks to the domestic plants and animals that greatly helped in this area. Focusing on farming and food production allowed the population to grow, and building up food surpluses meant that not everyone had to farm and that some members of the population could become specialists in crafts, thereby giving the opportunity for gifted people to invent and experiment. The large, dense populations with the necessary bureaucracy that resulted meant that ideas diffused easily in this area, and competition between different groups likewise helped spur Eurasian society on in its development. Thus it is in Chapter 18, where Diamond looks at the reasons why the Old World triumphed over the New World, he states:
Rivalling germs as proximate factors behind Europe's conquest of the Americas were the differences in all aspects of technology. These differences stemmed ultimately from Eurasia's much longer history of densely populated, economically specialised, politically centralised, interacting and competing societies dependent upon food production.
Here, we can see how food production, thanks to the availability of a greater number of domestic animals and plants, is thus related to the development of technology such as steel and literacy, establishing the causal link that has done so much to shape our world as we know it.
The basic answer here is that having domestic plants and animals (in other words, having an agricultural economy) allows a society to have a higher population density. When you have more people living in one place (and when many of them do not have to be engaged in getting food), you can have all the other things you mention.
Literacy comes around because densely populated places need governments and records. The governments are needed to control large populations and governments all need records for things like taxes.
Steel weapons come about because the people who do not have to be getting food can become craftsmen. Some might learn how to work with iron and steel.
Empire comes about because the agricultural societies will have lots of people equipped with good weapons and carrying germs that came from the domesticated animals.
So agriculture makes for more people, some of whom don't have to be engaged in getting food. This allows the three things that you mention. Because Eurasia had lots of potentially domesticable plants and animals, it got agriculture first.
According to Jared Diamond in Guns, Germs, and Steel, the availability of domestic plants and animals explains why empires, literacy, and steel weapons developed earliest in Eurasia because this availability led farming to develop in Eurasia before it developed anywhere else. Farming, Diamond says, allows empires, literacy, and steel weapons to arise.
For Diamond, farming is the key to power and wealth. Without it, a society remains at the hunter-gatherer stage and cannot become strong and rich. This is the case because only farming societies can build food surpluses and because farming societies are much more likely to be sedentary than hunter-gatherer societies.
A developed farming society can produce a great deal of food. Importantly, not all of its people need to be involved in obtaining food. This means that there are people in the society who are free to do other kinds of work. These people can invent and work with technology such as the technology needed to develop steel weapons. They can develop writing and keep written records. They can act as priests in a religion that supports their ruler and their society. They can act as soldiers. All of these things make it more likely that farming societies will develop literacy and steel weapons and will eventually gain empires.
So why does the availability of domestic plants and animals lead to farming? Diamond says that farming arises first in those areas where there are suitable plants and animals. People in such areas are much more likely to find plants to give them protein, oil, fiber (for cloth) and other things that they need. If there are fewer domesticable plants, they may only find a few of these things and farming will be harder to develop. People in such areas will be able to have animals that will help them in a variety of ways. The animals will provide protein. They will provide fertilizer in the form of their manure. They will provide power to pull plows and other devices that will make farming easier.
In these ways, the availability of domestic plants and animals (which Eurasia had more than any other continent) will lead to the development of farming which will, in turn, allow for the development of steel weapons, literacy, and empires.
Of course, Diamond spends a lot of words answering this, so this is going to be a very boiled-down summary...
The basic idea here is that the availability of domestic plants and animals allowed for a bigger, more concentrated population to rise up in Eurasia before it arose anywhere else.
Domesticated plants and animals allow people to start agriculture and agriculture allows relatively large numbers of people to live in one place fulltime. When people live like that, you start getting formal government with bureaucracies and you get more trade. Both trade and government can benefit from literacy. So agriculture leads to more people leads to government and business and that leads to the need for writing.
As far as steel weapons, having more people in one place allows some people to specialize in things like working with metal. Having "extra" people to work on things like that makes innovation more likely.
So the short answer is that domestication led to higher population densities which led to literacy and steel weapons.