Humankind continued the "space race" well into the 21st century. That's not to say it's any different from what's happened IRL, as I don't necessarily want my world to be an alternate world rather than a possible future. Countries and private companies that are financially able join the race, so basically humanity is propelled into the cosmos by sheer determination, curiosity, and probably greed.
Cut to an appropriate amount of time later: Generations of humans live and die in orbital colonies populated by the overflow of people from Earth and their children's children etc. People even live around the many moons of the outer planets, surviving off synthesized and raw materials mined from the gas giants and asteroids by a robust commerce system. Space travel is still difficult, but commonplace. I imagine outer Solar civilization to be mostly cut off from the inner homeworld(s) due to the immense distances that take years to cross.