Rather than further broaden their diet, they become food producers. Now, rather than animals and plants only as food, we begin to see them focusing on other products. For example, the live animal can produce milk, wool, provide traction. The Neolithic is characterized by plant and animal domestication. They just described taming, not domestication.
Domestication occurs when humans begin to control the reproduction of plants and animals.
Once we begin to chose which plants and animals get to reproduce, then domestication occurs. Realize that there will be a lag of a number of reproductive cycles, generations, before we see this in the animal or plant, when this change occurs, the reproduction of the plant or animal species in question has already been under human control for a long time. Plants lose their ability to distribute their own seeds. Think of a dandelion. All it takes is a puff of wind and the seeds are distributed.