In Chapter 1 of Guns, Germs, and Steel, Jared Diamond notes that there is some controversy over this issue. He says that not all scholars agree on what, exactly, happened to the megafauna of these two areas. However, Diamond argues that the megafauna were killed off by human beings.
It is indisputable that the megafauna of these two areas died out at specific times. On p. 42 of the paperback edition of the book, Diamond tells us that there is no fossil evidence of megafauna in Australia in the last 35,000 years. On p. 46, he tells us that the American megafauna became extinct somewhere between 17,000 and 12,000 years ago.
What is less clear is why the megafauna died out. There are some that believe that it was climatic changes that led to the extinctions of these animals. However, Diamond clearly says that he believes the megafauna were killed by human beings. As he says on p. 43
Personally, I can’t fathom why Australia’s giants should have survived innumerable droughts in their tens of millions of years of Australian history, and then have chosen to drop dead … when the first humans arrived.
In short then, the megafauna in Australia and the Americas went extinct soon after human beings arrived on those continents. Diamond believes that the human beings caused the extinctions.