Heaving read about the Voluntary Human Extinction Movement before, I'm not sure that they really think that this will happen.
It's sort of like religion is supposed to work, like an Idealism. I think the idea is that we try to not pass on our DNA, we try to minimize our use of everything, and our impact on the planet.
Not everyone is going to, so not everyone is going to die, or go extinct. It also enumerates many things that one can do to ease the burden on the environment.
The problem is, there ARE too many people for the planet, and the limited resources, and said resources are very poorly distributed.
It's the snowball effect, like wankstaboi mentioned. People are living longer, due to advances in medical care, infant mortality is less, Technology has enabled people to live in harsher environments, and all of that has led to a massive increase in people.
However, the environment is trashed, there's all sorts of chemicals in the water and atmosphere. There are several gigantic garbage zones in each of the world's oceans, and in many places, virtually all the seabirds have some plastic in their stomachs.
Sustainable production is almost non existent, and many things that are either produced green, or gown organically are outrageously expensive. $20 per pound of Grass fed, organic Beef? Hell NO! And why should it be $20 per pound? I dont' get that.
Basically, we need to rethink our role in this planet, and what we are doing for the future, because, quite frankly, in 100 years, this world may not be here.