The opinion of the archaeologists that the mammoth has been killed at Domebo is based on two things:
Weapons; on the site where the mammoth was found the archaeologists also found 4 projectile points, which had the purpose of killing big game animals, and they had markings that suggested that they were used.
Bone placement; the placement of the bones of the mammoth suggested that they were placed in a manner which was used when the flesh of the animals was sliced from their bones.
The answer is - hunter-gatherers.
The Archaic Indians in North America were hunter-gatherers. They were reliant on hunting animals and gathering plants and their fruits and vegetables for survival. Still, in the early period until the landscape changed its appearance after the end of the ice age, they were mostly focused on hunting, and they were usually hunting the big game animals, including the bison, and also the mammoth, wild horses, giant sloth, before they went extinct. This was due to the living conditions in that period, and that the agriculture was still not taking place in their societies.