Barring genocide, the solar system becomes crowded

In the mundane scenario, there is no FTL drive and STL drives are severely limited by available energy or mass ratio or both. Any starships are likely to be robotic, or perhaps carry a very small crew in hibernation (or more likely dead and stored; rebuild them at the far end). So eventually either population pressure or economics make local space colonization attractive on more than a "look, we did it!" basis. It might take a thousand years, though.

In the slightly-exotic scenario, there's no FTL but lots of energy or some other gee-whiz factor enables asteroid cities and comet mining. The solar system gets colonized relatively quickly. Resources are contested. Eventually some of the asteroid cities stock up on spare parts or spare nanoassembly factories and fusion plants and head out for other systems. (Maybe.)

In the exotic scenario, FTL comes along. Other systems with Earth-like planets are preferentially colonized, but unless you also grant an unlikely ground-to-orbit lift capacity, no significant part of Earth's population gets to leave. It might slow down growth, but it won't stop it. So Earth gets crowded, and see #1 or #2.

Or maybe we upload into cyberheavens run on computronium assembled by von Neumann replicators out of scrap material like Manhattan and Tokyo and Jupiter, and the Matrioshka brains build a Dyson swarm, which then gets crowded...

Barring genocide, the solar system becomes crowded

In the mundane scenario, there is no FTL drive and STL drives are severely limited by available energy or mass ratio or both. Any starships are likely to be robotic, or perhaps carry a very small crew in hibernation (or more likely dead and stored; rebuild them at the far end). So eventually either population pressure or economics make local space colonization attractive on more than a "look, we did it!" basis. It might take a thousand years, though.

In the slightly-exotic scenario, there's no FTL but lots of energy or some other gee-whiz factor enables asteroid cities and comet mining. The solar system gets colonized relatively quickly. Resources are contested. Eventually some of the asteroid cities stock up on spare parts or spare nanoassembly factories and fusion plants and head out for other systems. (Maybe.)

In the exotic scenario, FTL comes along. Other systems with Earth-like planets are preferentially colonized, but unless you also grant an unlikely ground-to-orbit lift capacity, no significant part of Earth's population gets to leave. It might slow down growth, but it won't stop it. So Earth gets crowded, and see #1 or #2.

Or maybe we upload into cyberheavens run on computronium assembled by von Neumann replicators out of scrap material like Manhattan and Tokyo and Jupiter, and the Matrioshka brains build a Dyson swarm, which then gets crowded...