To be fair, it's very much up for debate how much of a part humans played in the extinction of mammoths and other Pleistocene megafauna. There isn't a whole lot of evidence to show that human activity drove 'em to extinction.
It's common consensus that humans were the primary cause of the Pleistocene Extinction Event in Australia, at least. Hunting, firestick farming, and other activities caused the demise of almost entire ecosystems. Such animals include the Giant Short Faced Kangaroo (Procoptodon goliah), the Genyorn (Genyornis newtoni), the Rhinoceros Wombat (Diprotodon optatum), and the Pouch Lion (Thylacoleo carnifex) among many others, ending with the Thylacine in recent times. Any drastic climate change occurred far after the Australian decline began.
In addition, it's not crazy to believe that humans played a key part in not only the Pleistocene Extinction Event globally, but also perhaps the end of the Paleolithic Ice Age entirely.
Think about it. Hunters take advantage of the massive diversity and unthinkable density of megafauna in the frozen north, along the steppes. As human populations go up, our prey's goes down. Less megafauna on the cold grasslands due to pressure from humans means the forests can encroach back onto the open plains, accompanied by tundra.
Mammoths were among these megafauna. Huge proboscideans like themselves are the very engineers of their environment, making them a driving influence in the landscape around them. These northern Pleistocene megafauna, mammoths especially, inadvertently culture an environment very different from what we know today.
Their influence is absolutely
massive. We know how much even small herds of large herbivores can do when returned to the tundra and taiga, thanks to Pleistocene Park. The grasslands of old, slowly but surely, begin to return.
These aforementioned grasslands are able to insulate the permafrost underneath. Forests and tundra, however, are much worse at that job. This causes greenhouse gases trapped for millennia before to be released back into our atmosphere.
See what I'm getting at here? People who deny ancient humans' roles in the extinction of species often blame climate change. But what if us humans were the culprits behind much of that, too? What if we kicked the domino and started something much larger than we thought possible?
What seem to just have been dead-end extinctions may have kick-started the Pleistocene Extinction event around the world.