It's been happening for a very long time. Some put it back to the times when humans left Africa. It's known as the Holocene Extinction [0]. Though, it has never been as bad as these last couple of centuries.

I think the fact that people don't seem to be aware of this is a massive failure of our society and in particular the education system. I do not know, but I would like to believe that it would have made a difference if we were all aware of how huge our impact is on the biosphere is.

Refs: [0]: https://en.wikipedia.org/wiki/Holocene_extinction