I am completely bewildered by Influence (not the use of influence on cities/outposts and wonders), but its influence on different cities and other civs. What does it do? It's not explained anywhere. I'm an Aesthete and I decide to "dominate" a territory with my influence. What does that do? 

All I can tell so far from my own playing is that influence can get us more money if we have control over a territory. I too chose a side that had Aesthete so I saw the same options as you. I have no idea where it shows how much money we can make though from influence, I just know we get money lol. If we can steal land by having dominant influence over it, or for that matter do anything other than having money, I have no idea. I don't seem to even incur relations problems by having dominant influence over others afaik. 

I think the influence system isn't fully fleshed out yet, or at least not the UI for it.

At a certain point in the game, it becomes a grievance when an AI controls a territory where your culture is dominant, so you can use that as a reason to go to war and take that territory.