After the Earth was used up, we found a new solar system and hundreds of new Earths were terraformed and colonized. The central planets formed the Alliance and decided all the planets had to join under their rule. There was some disagreement on that point. After the War, many of the Independents who had fought and lost drifted to the edges of the system, far from Alliance control. Out here, people struggled to get by with the most basic technologies; a ship would bring you work, a gun would help you keep it. A captain's goal was simple: find a crew, find a job, keep flying.
Thank you for the fine quote from Firefly. This was the single most important storyline in the history of drama. It depicted free people living lives of integrity outside the structure of Big Brother. No wonder it was cancelled.