I personally don’t think hunting was even that significant, but for some reason it gets all the attention. I like to compare humans to so-called invasive species (because we are one); kudzu vine doesn’t hunt and yet is considered a nasty invasive. Rabbits are herbivores, but they are invasive in Australia.
Those invasive species cause ecological destruction (there are some people who somewhat dispute that though, but this comment would be long) just by competition and more subtle ecological interactions.
That’s probably how humans did it, “it” being the extinction of over half of most continents’ megafauna, having tools and technologies essentially makes us a super-invasive, the most efficient invasive species ever.
By the way I can never get tired of stressing that we do know that humans caused the Late Pleistocene extinctions (which is a massive rabbit hole worth a Wikipedia binge), notwithstanding any debate. I don’t know how there even is any! Over a dozen times across tens of thousands of years, on large landmasses around the world, human arrival coincides exactly with local extinctions. That isn’t an accident.