OK i will try to answer this for you:-
Influence bases do NOT boost the influence of planets in their ring - they generate their own influence, and the modules added to them boost the base influence they generate. Also, ignore the area of effect of these bases; the effect tapers off with distance, so for best effect place them touching the planets you want to influence.
Influence is a measure of how much impact your society as a whole has on the galaxy. With a high enough value you can start to take over other empire's planets without the need to send in troops. Also if you can get 75% of the galaxy under your influence while controlling 75% of the assets on the map you win a Cultural Victory. There are a few ways to increase your influence over a given area. The most obvious is if you have a planet in the region of space. The population on that planet produces influence. The larger the population the more base influence you produce. You can increase the planetary influence by building an Embassy or Cultural Exchange Center on the planet. To extend your influence into areas where you may not have a planet to colonize you must build influence starbases. These multiply your influence in their sphere of influence by the bonus the modules on the starbase provide. No need for one of your planets. So lets say you've got a couple of colonies from another empire within your borders. If you have at least a LOW espionage rating with the empire you can see what the influence that planet is producing and also what value above that you are producing. If you achieve a value of 4.00+ then your influence will start to sway the population to revolt and join your empire. Influence starbases that have these planets within their sphere of influence can help to really increase the likelihood of them defecting. If another empire is threatening one of your planets you can counter their influence in the same manner that you use to increase yours. This is why there are times the AI will build a lot of farms and/or embassies on a planet. It is trying to keep the planet from defecting.
Influence + 10 = Your influence can cause other systems to defect and join your empire...or you can lose your planets to another, more influential government. It's basically an "idea war."
Hope that helps some
