Mammoths lived during the Pleistocene (about 1.8 million to ~10,000 years ago), when the most recent ice ages took place.  During the Pleistocene, the Earth went through a cooling period and glaciers covered large areas of land.  There would also be short warming periods when glaciers retreated.  Most ice ages had cycles of long cold periods and short warming periods.  Mammoths were adapted to live in these colder climates and lived in open prairies and the tundra and taiga regions of the world.