Stellaris Influence Guide (4.5): How to Get More and What to Spend It On
Influence in Stellaris is the political resource that pays for new systems, and you raise it through a short list of sources: a flat +3 a month, up to +2 from Power Projection once your navy is big enough for your empire, +0.5 for every rival, and later your ethics, council traits and the Will to Power decree.