I'm fine with the idea of a species -- or whatever definition you want to use here -- expanding beyond one solar system. But without at least faster-than-light communication, I don't think it would be possible to maintain a cohesive civilization at that scale. And without faster-than-light travel, you're also going to have things like species rings developing.
Whether a civilization expand beyond their solar system is entirely depending on if they develop the technology to do so in a relatively short time compare to their life span.
One good reason could be habitable planets. There may not be enough habitable planets left in the solar system.
Extra: some sci-fi look at Kardashev scale https://en.wikipedia.org/wiki/Kardashev_scale
Escaping one solar system with some amount of colonists is probably harder than developing the technology to not need whatever counts as "habitable planet" at the other end.
They'd use the new system's ressources for all sorts of things, least of which would be biological survival.