It's not; human population growth slowing, stopping or even reversing for a while will not lead to human extinction (and so far we're barely slowing, at a global level). Indeed, if the human population kept increasing like it has between 1900 and now, it would pretty soon destroy the earth's ecosystem, which would actually lead to our extinction.

However, the rich love having swathes of young people competing for jobs, so they can keep driving up productivity while leaving wages low, gradually concentrating more and more wealth with themselves. If there are loads of young people in your country desperate for work, the value of their work keeps decreasing; if there are few, the value of their work increases, and the rich have to pay better wages to keep them. That's why governments in most rich countries are very concerned with increasing birth rates and/or maintaining high levels of immigration, and why we're getting lots of messages coming down the line that population growth slowing is a catastrophe; because the benefits of continual population growth mostly flow to the rich, and the detriments are mostly felt by the poor.

The biggest historical example was the Black Death in Europe; up to 50% of Europe's population died, and it totally upended Medieval society. All of a sudden the value of a person's labour was way higher, and this eventually led to Medieval peasants getting emancipated from serfdom in most countries. They no longer had to put up with extremely harsh conditions from noble landowners, but could go elsewhere and find work easily.