Hey Demented Cargo, so Influence doesn't really affect anything except a few subtle ways.
One, if you get 10k influence you will get access to the Exclusive Access forums which gets you access to monthly Q&As with the developers.
Two, it's used frequently to determine order when reservations are made. So when title holders pick during Domain and Settlement Selection, the people with the highest influence will get first pick in their title tier, likewise when surnames were reserved the higher influence folk got first pick as well.
There are a couple other little things like being able to add a forum avatar or add links to forum posts if you hit at least 250 influence, but that's pretty much it.
From the DM you sent me it sounds like you got the Proprietor package, so not a lot of this will affect you unless you get up to 10k influence, or something else comes out that potentially is gated by influence, like maybe heraldry creation?