How do influence points correlate with the actual territory you occupy?

I know that as influence points increase, your territory expands, but if I get extra influence points by trading with other civs, does it contribute to territory increase?  Also, how does influence starbase affect all of this?

You have this backwards. Territory you own gives you IP, IP can't give you territory.

Technically, it's not even territory that gives you IP, it's the total influence output of your civilization. On a very dense map, you may have a lot of influence output, but actually own a small fraction of the map - one test game I did I had ~95% of the influence generated on the map, but only owned 35% of the map. On a very evenly distributed map, you may own 75% of the map but only generate 30% of the influence.

IP is the total influence output of your civ each turn, piled up until the UP vote each year. By trading IP, you could *thoretically* gain enough votes to swing the UP resolution the way you want. Unfortunately, the number of IP votes in play makes it practically impossible for such a situation to occur. Even on moderately sized maps, there may be 100+ planets (400+ in immense games) and millions (or even tens of millions) of IP votes in play. There is no practical way to buy enough votes to swing a UP decision, unless you are so close to having a majority already that buying the votes is unnecessary anyway. In very small games it may be possible to buy enough votes to matter, but then the money supply available would likely not be sufficient to buy enough votes.

Influence starbases contribute to your total influence output, on the same scale as a 2-4 billion planet if the base is fully built up.