Serious question. What does "+25% influence change" in a controlled/exploited system actually means?
Hard to quantify, given how shrouded in mystery all the influence and reputation stuff is, but my understanding is:
If you complete a mission to, say, transport some food in, that will earn a given amount of influence for the minor faction you complete it for. Let's call it 100 influence for simplicity. With that bonus, they'd earn 125 influence.
As a result, that faction will find it easier to become the controlling faction in the system.
That is, assuming you're not just taking missions from every minor faction in the system, in which case it'll pretty much even out.