One possibility: It is a weird case of "this moment in time is special because I am alive".
Humans have been around for thousands of years but I am alive now. I cannot think of what the world could be like without me so therefore the human race must end in my lifetime.
This in not a new occurrence either, there have been "end of the world" groups in almost every generation in history.
It is worth noting that people always think the apocalyptic event will happen within their lifetime.
The only real difference now is that the end of the world is now caused by humans via nukes or destruction of the environment rather than supernatural forces.