I can give a very cursory answer, but you might get a much better answer by cross posting in /r/askanthropology.
Hunting deer doesn't require a lot of equipment - it is not so difficult even for just an individual with a bow and arrows. Larger game, like mammoth, were probably group efforts. We also have evidence of other tactics people would use, such as bison jumps on the Great Plains. You find a suitably high cliff or other drop and chase a part of a herd of bison off of it. You then harvest what you need from the dead and wounded bison. Just generally, people are very creative in hunting large game.
The Atlatl was mainly used by Stone Age people to hunt big game. It allowed someone to propel a spear at speeds much higher than can be achieved by simply throwing it. It was also quite accurate and had a long range, allowing humans to stay at a safer distance while hunting.