That is the conclusion reached by Brian Huntley and colleagues of Durham University, UK, as reported by BBC (sorry but I cannot find a more direct reference at this time).
According to the British scientists, after running several models for the extinction of mammoth and other megafauna of the Ice Age, the model that is most plausible is that they went extinct because of loss of grasslands and not human hunters' pressure.
During the height of the ice age, mammoths and other large herbivores would have had more food to eat. But as we shifted into the post-glacial stage, trees gradually displaced those herbaceous ecosystems and that much reduced their grazing area.
Update (Aug 19):
Found the relevant paper:
Judy R. M. Allen et al., Last glacial vegetation of northern Eurasia. Quaternary Science Reviews, 2010. Pay per view.
There is also an alternative news article at Science Daily (typically a literal transcription from the press release) that uses maybe less absolute terms than the BBC one:
The change from productive grasslands across large areas of northern Eurasia, Alaska and Yukon to less productive tundra-like habitats had a huge effect on many species, particularly on the large herbivores like the woolly rhinoceros and woolly mammoth. Mammoths and other mega-mammals found it increasingly difficult to find food. We believe that the loss of food supplies from productive grasslands was the major contributing factor to the extinction of these mega-mammals.