There’s an interesting debate at Marginal Revolution over the patterns of extinction of big game around the world as humans show up over the last 100,000 years.
Here’s my super-stylized (i.e., not terribly well-informed) model:
That’s my best guess. Maybe something else like a newly introduced disease wiped out the local megafauna. It’s also possible that climate change, perhaps induced by newcomer humans setting wildfires, whether accidentally for their own use or intentionally to alter the ecosystem, had large consequences for megafauna extinctions. But my best guess is: the Indians ate them.