What people often forget

Yeah but if nature didn't had people, there wouldn't be any tv shows. Or books. Or good food.

But, we're a product of Nature. So in that sense, we are Nature.

and so do animals

When people say 'Save the Planet' what they're really referencing is save humankind. If we eventually wipe ourselves out through war or disease, Planet Earth will continue on quite nicely without us like it did for billions of years before we came along.

More like nature doesn't want people after people destroying it.