Prehistoric hunter-gatherer societies had not yet developed specialization of food production, either by herding animals or planting crops. Their lifestyle was one of subsistence, basically "feast or famine." They did travel from place to place, often following game animals and frequenting areas where roots, berries, etc. would be seasonal. In essence, their lifestyle was one of necessity rather than preference. They did not have the tools necessary to practice agriculture or animal husbandry; rather they only had crude weapons for hunting and crude tools for processing food. At such time as settled agriculture was developed, they were able to produce enough food for themselves in one place. This not only allowed them to give up previous lifestyle; it allowed for the production of more food than was needed. Overproduction of foodstuffs allowed specialization to develop, and ultimately class distinction.
The main reason for this is that the hunter-gatherer lifestyle is not one that is compatible with being sedentary. It is not really possible to stay in one place all year round and still maintain that lifestyle.
Hunter-gatherers have to follow the food sources. For example, when herds of their prey migrate, they need to follow those herds. As another example, different kinds of plants mature at different times of year. Hunter-gatherer bands often need to move around so as to be able to gather these different food sources as they mature.
So hunter-gatherers, for the most part, are not able to stay in one place because their food sources do not stay in one place. For this reason, they tend not to have permanent homes or dwellings.