Domesticated animals can help with the production of food in a number of ways. These ways are described, among other places, in Chapter 4 of Guns, Germs, and Steel. There, Diamond tells us that the possession of domesticated animals helped people in four ways.
First, animals provided meant. Second, they provided milk. Both of these are very important for feeding people. It is important for people to get protein if they are to be healthy and vigorous. If people have domesticated animals, they can get protein from their meat and, in some cases, from their milk. If people are healthier, they can work harder and they can have more children. These things make populations and civilizations grow.
Third, animals provide manure to fertilize crops. Fourth, they can pull plows. Both of these things allow people to farm more efficiently. When they farm more efficiently, they can feed more people. When farmers can feed more people, more specialists can do things like creating technology or passing down knowledge to one another.
As Diamond says on p. 89 in the paperback edition,
All of those are direct ways in which … animal domestication led to denser human populations by yielding more food than did the hunter-gatherer lifestyle.
Denser populations make for a more rapid growth of civilization.