Once you've colonised the star system during its close passage to your own, it will continue its journey and eventually pass nearby other star systems, which can then be colonised as well. So every colony will spawn more colonies, leading to exponential growth.

At least I think that's what they meant. It's an interesting idea for sure.

Yes, that's exactly what I meant.

I think they mean that if you start in solar system A, when solar system B flies by, you enter its gravity and colonize the system. Eventually solar system B will fly by solar system C, where you can repeat the process.

B-Arc