1. People want to go into the solar system. They need something to do.
2. We're nowhere near the level of smart, self-repairing automation. Given the distances involved, there's value in having a community of smart, agile, self-replicating flesh-and-blood humans to tackle unknown engineering challenges as they appear.
3. People keep expanding, and settle for good. Years pass. An interplanetary economy forms. People on the further edges focus on resource extraction and building up infrastructure for long-term settlement - but they still need things only Earth can provide, like food (and organics in general). So they trade.
4. Decades pass. On the outer edges of humanity, people are born who've never seen Earth. They start to develop their own culture and identity. Mars is being colonized. The trade arrangements is stable - Earth sells organics and R&D, they buy lots of resources to use for in-orbit manufacturing.
5. Many more decades pass. Mars is full of people who were born there, and never seen Earth in the first place. They trade with Earth and the space dwellers - the Belters, in Expanse - for resources. They develop their own culture too. Over generations, the Belters become adapted to low-gravity conditions, and can no longer survive Earth's gravity - cementing the cultural divide.
6. More decades pass, you arrive at the world of The Expanse. Mars and the Belt both declared independence, as they're their own separate cultures and nations. Trade dependencies, however, remain. Earth and Mars both have the advantage of infrastructure density and crucial resources abundance - they're developing their planets and surrounding orbital space. Stuck in an arms race, they develop superior military. Meanwhile, the Belters are stuck in a disadvantaged position. They own most resources in the Solar System, but can't effectively control them with inferior technology. And none of the resources are actually missing on Earth and Mars - they're just more expensive to get. The Belt has all the riches, but no leverage to turn them into wealth and power.
That's how you arrive at "mining asteroids in poor third world-like conditions".