Influence is a score attached to every post that goes up as people engage with it — the more activity a post gets, the higher its Influence climbs. The feed isn't purely chronological: posts with higher Influence are prioritized and shown higher up, with newer posts breaking ties among posts with similar Influence.
Tapping the Influence number shown on a post opens a short explanation of what it means, in case you're seeing it for the first time and aren't sure what the number represents.