Sir, u just need to read attentively what influence option does. Your existing colonies send % of their food production to your outposts with food transports (automated ships). U can see which colony provides food ships for the outpost in the outpost screen and can change it there. What influence option does - it increases % of food production sent to outpost by 100%. So if u check the system that is providing food for outpost after influence option is activated u'll be able to see all the changes it does.

Hope this helps.

I mean, sir, i already told u what it does. And tbh i have nothing more to add to my explanation. I'm not sure what u're missing exactly but my wild guess is that u expect influence option work just like the dust one but it's completely different mechanic, that unlike dust option doesn't grant it's (visible)effect right away. First u need to wait for the (food)ship to be sent from colony(it ain't sent right away either - u can see turns till it'll be sent in outpost view) and then u have to wait till it arrives to outpost(and if colony providing food ships is far from outpost it might take a while) and only after that u'll be able to see the change in outpost growth time (not it's food production).

EDIT#1: I took some time and applied my incredible paint.net skills and made this - https://imgur.com/a/FAF8Y hope it'll help u to sort it out.

EDIT#2: Actually i was mistaken thinking(for some reason :D) that it makes 100% of your food production go to outpost, while it just doubles amount of food sent from colony to outpost. I edited my first post to correct myself there.