You don't need to have a ship in orbit to do this last colonization step, and it costs no resources (critically unlike other factions, you do not consume your ship). Once a system has been vined, you can instantly colonize a planet (that you have the colonization tech for) on it immediately and without any cost. Starting from their home system, these 'vines' are able to extend to nearby systems this works by performing a special action to 'add' a new system to the network of vines once it's added, if any nearby systems are in range, a 'vine' will extend out to them from every nearby system that's also vined.
In order to colonize planets, they need to create a special entity viewable on the galactic map, called a Celestial Vine.
The Unfallen completely replace the normal colonization mechanic. They're described in detail in the Traits tab. These are a special population type only accessible via curiosity exploration.