Why might it be important for an atheist to feel, as Sire states, "Existentialism's major interest is in our humanity and how we can be significant in an otherwise insignificant world."
The Universe Next Door: A Basic Worldview Catalog by James W. Sire
The main idea of existentialism is that "existence precedes essence." That is like saying that you are born as a tabula rasa, a "blank slate." There is no Divine plan and no preconceived point or reason why we are here. Therefore, we are given the freedom and the responsibility of creating ourselves (mentally, culturally, etc.) and the world. With no divine or moral script, we are free to choose. With that freedom comes great responsibility. Under this existential belief system, if the world is a horrible place, it is our fault. It is necessarily up to us to make ourselves and the world a moral and ethical place.
This insistence on personal (and social) freedom and responsibility puts the burden (but also the honor) of creating the world on people's shoulders. In Existentialism is a Humanism, Jean Paul Sartre writes:
When we say that man chooses himself, not only do we mean that each of us must choose himself, but also that in choosing himself, he is choosing for all men. In fact, in creating the man each of us wills ourselves to be, there is not a single one of our actions that does not at the same time create an image of man as we think he ought to be.
It is important for an atheist (or a religious person for that matter) to embrace the honor and the burden of being responsible for ourselves and the world.
First, we must recognize that it is important to realize that different atheists will have different attitudes and points of view just as religious people are not all the same. This means any answer to this question is a generalization.
That said, we can argue that an atheist would feel this way because they do not believe in God. For believers, there is no question that humanity and human life are significant. People and their lives are significant because they are created by God and because it is important for them to live up to what God expects from them. But atheists have no such built-in source of significance. They may well feel that their lives are meaningless or insignificant. Therefore, they will need to seek philosophical approaches that can imbue their lives with meaning.