How did nomadism eventually lead to domestication of animals and plants in the development of civilization?
First, we must note that nomadism did not actually cause plant and animal domestication (also called agriculture). In other words, it was not necessary for people to be nomadic in order for them to end up domesticating plants and animals. Instead, people who were nomadic gradually and unintentionally did things that ended up allowing agriculture to develop.
Nomads are typically hunter-gatherers. They get their food by hunting animals and by gathering edible plants. As they gathered the edible plants, they unintentionally made agriculture possible. They gathered the best plants and brought them to their camps to eat. As they did so, the seeds of those plants would be deposited (when they threw the seeds away after eating the meat of the fruit, or when they excreted the seeds after eating) in concentrated areas. Gradually, this led to relatively dense areas of plants that had been selected by the human beings. Eventually, people realized that they could intentionally plant these plants and agriculture began.
The same type of thing happened with animals. We do not know exactly how it happened, but we believe that people selected the animals that were most helpful to them (the tamest animals, for example) and kept them near. Eventually, the tamer animals interbred and produced still tamer offspring. This led to truly domesticated animals.
Thus, agriculture originated because of things that nomads did. Nomadism was not necessary to the development of agriculture, but the actions of the nomads led to that development.