I couple of in-games messages have appearedwhile playing, stating that my city/outpost is now in the sphere of influenc of another civilization, usually a neighboring one.

What does this mean?  I cannot find any mention of its effects in the encyclopedia or tutorials.  

I'm finding that more extensive descriptions of many terms in the game is lacking, but this is the most obvious one to me.

Hey, so you can find the Sphere of Influence when clicking on the Society button 

Agree that sometimes it is complicated to get certain aspects of the game.

yes, but I don't see what it *does*

The main effect is on producing influence and gaining support for the war. Each populated territory generates a certain amount of influence points per turn, most of which go to the nation with the most influence in the territory. So if your territories are in another nation's sphere of influence, you are gaining less influence points, as well as claims claiming those territories and occasionally demands and wars can arise.