I'm completely baffled by it, and yes, I searched these forums and cannot find any comprehensive explanation of what this is or what's the point.

I agree. Other than using it to build outposts and attach districts it doesn't really seem to have a point? I had so much influence by the end of my playthrough and I couldn't spend it on anything and I was kind of like what's the point of it?

Could someone help me understand how to gain more influence? I just started playing the open dev for the first, and my neighbor manages to build several outposts way before me and then I get the message that my first city will be converted by them because of my neighbor's influence. But I think I only got like 10 influence when I moved to the ancient era and have 0 new influence per turn. How do you get it?

If you have a City, it is not possible to gain below 3 Influence per turn.

Did you even "Upgrade" your first Outpost to a City after reaching Ancient Era?

There isn't much to explain there. It's a resource, like money.
Spend on building outposts/cities; linking outposts to cities; claiming wonders; absorbing cities into each other.

You will also notice very quickly that each new outpost increases in cost and that outposts are cheaper to build next to a city.

Influence is easy to understand as a currency, but it also seems to affect what culture each region is, which does... something? Nothing? Just help you to get Oppressing My People grievances?

In theory this could be really interesting and cool but it doesn't seem fleshed out yet.

Yeah, I'm trying to learn it as well. I made the mistake of building a ton of outposts, not realizing influence was a finite resource I was spending. I have only 1 city and all my territories are dominated by the influence of other civs now. I'll probably just learn it through trial and error, but yeah I was looking for some kind of detailed explanation of it somewhere in game. It seems pretty cool so far though - it made my citizens demand I choose the same civic policy of my neighbor which is both realistic and interesting as a game mechanic.