On the Galaxy map, you will find these artificial celestial bodies run by pirates. Each Facility entails some special bonus for a corporation that manages to seize it. Facilities grant bonuses to a corporation only if located on its territory, with pirate presence cleared from the said Facility.

How to? Easy: once you build a Dyson Sphere or an Outpost nearby, you can launch a rally to get rid of the pirates based in the Facility.
Once seized, the Support Facility starts awarding bonuses to corporation employees whose bases are located within this territory.

Support Facilities range from level 1 to 5.
The closer they are to the center of the Galaxy, the higher the level, and the better the bonuses you want to fight for!
Nevertheless, pick Facilities wisely: identical bonuses do not sum up!

In addition to bonuses, the corporation will receive passive hourly income in corporate credits from every Support Facility seized. This income does not depend on the level of the Facility, but serves as a great motivation to conquer and expand the territory!