I'll do my best I am not a math guy so I can't give you the formulas that drive it.
You start with your home world which has an influence bonus and a growth factor. Each planet you colonize adds to your influence.
You increase your influence by researching influence techs which opens up influence improvements that can be built on you planet. The basic influence improvement is the consulate. This should be built on every planet if you are interested in spreading your culture. It will eventually upgrade to an even stronger improvement as you research the techs.
The ideology tech also give you access to powerful influence improvements. Well, at least the beneficent choices do.
You can also build culture rings on any star base that doesn't have another type ring. For instance any star base that is only relic research can also have a culture ring. You can also build culture star bases at random or near the AI if you wish. One of the ideology choices make all you star bases generate culture regardless of their purpose.
Influence effects have been nerfed to the point that they are relatively weak imo. but you can still overcome the AI with culture on the smaller maps. You could probably do the same on an insane map but it would take at least 500-600 turns maybe a lot more. I haven't heard anyone claiming an influence victory on an insane map.
To try to be more specific to your questions,
Does it spread purely X tiles from the generator? Or does it add it's influence growth somehow to your total border regardless?
It works for your entire empires ZOI.
if I have strong influence generators on my empire's borders, does it help my influence at all to "backfill" with cultural starbases in the inner regions of my empire?
There is a local effect for influence generators, but the strongest effect is the overall influence generated by all your colonies and star bases. When you get a home planet bonus which increases it's already strong influence you can actually see your influence expand. Later when you border other factions, if you have been growing influence you can see on nearly every turn how you are encroaching and encompassing them. The starbases will of sourse help but they are not necessary. If you played GCII, the cultural starbases were very powerful locally. Not so much in GCIII.
Does it help at all to, say, build a couple of influence buildings on random inner planets, or am I better off making one "pure" influence planet and focusing only on making influence on border worlds?
Every influence building helps and I try to have at least 1 on every planet. I seldom have a planet devoted exclusively to influence but I imagine it would be powerful but overall you need your influence spread far and wide. A planet with 10 influence producers is no better that 10 planets with one.