The origin of that dust and gas is where the starstuff concept comes into play. Much of that Solar System-forming material was hydrogen and helium, which have both been around since the earliest days of the Universe. But abundant heavier elements — carbon, oxygen, nitrogen, iron, etc — first came into being about a hundred million years after the Big Bang, when clouds of hydrogen and helium collapsed into the first stars.
In the cores of stars, intense pressure and heat fuse hydrogen atoms together to make helium — the same process that powers our Sun today. When hydrogen in the core is used up, some stars begin fusing helium into heavier elements, like carbon and oxygen. These are the elements that make up most of what we are. As some stars get older, they go through periods where they throw off some of these elements off into space in huge stellar winds.
When very massive stars reach the end of their lives, they explode as supernovae. In those last moments, immense energy powers the fusion of even heavier elements like cobalt and nickel. As the star explodes, these elements are flung out into space. Together with heavy elements created through other stellar processes, they can eventually become ingredients for forming a planet, enriching the soil, feeding a plant, and becoming part of a human.
This is part of what makes life so fascinating. The actual material that makes up living things is basic — just chemical building blocks made by fusing heavier and heavier elements. But what life does with those materials is utterly magical.
What Carl Sagan tapped into is the wondrous nature of the fact that elements fused in stars can come together to form a person who can then contemplate those very stars. We are starstuff studying the stars — a way for the Universe to know itself.