The number in the brackets is the ratio of your influence on that planet to the next highest foreign influence on it. If that gets above 4.0, there is a chance your planet could switch sides to that empire. This is commonly called "flipping".
As far as I know, there is no definitive influence tutorial. I looked into it at one point, and influence is just a god-forsaken disaster of interconnected factors and unexplainable components that no one will ever completely solve it without cracking the game open and reading the code directly. We can give you workable estimates, rules of thumb, and generalities though.
In general, increasing your population and influence ability will increase your influence output - but not in any sort of linear manner. Your total empire population seems to have an impact on the influence output of each individual planet.
Each planet and starbase can be treated as a point source of influence, which then decreases with distance from the source. I call this an influence field, just so I can talk about it in terms of field strength. I've watched too much Star Trek, apparently.
The influence field strength at any diven point is apparently determined by taking all sources of influence, calculating what the effective strength at that point is for each individual source, then summing those. Your border is drawn where that field strength is 1 (when bordering empty space) or when the ratio of your field strength compared to the next highest empire's field strength reaches 1.
Influence field strength from planets and starbases decrease at different rates. Planetary influence decreases at about a linear rate with distance, while influence bases decrease with distance squared. As a result, if you are building an influence starbase to try to flip a planet, build it touching the planet. If you're trying to flip two planets with one base, don't put it exactly in the middle between them, build it touching one planet on the side nearest the second planet.
Influence starbases do NOT increase the influence output of anything in their area of effect - in fact, you can simply ignore that ring when planning where to put your base. The modules you add to the base increase the *base's* influence output, so they are still quite worth building.
When you get the influence ratio above 4 (the number in brackets you asked about) that planet has a random chance of flipping each turn. The Loyalty ability apparently decreases the probability of a threatened planet flipping on any given turn, and a propaganda center structure will make flipping a planet impossible, regardless of influence ratio.
Once you get past 4, increasing that ratio doesn't help. I've had a planet sit for an entire game year at well over 80 without flipping.
The Mind Control Center (only available to some tech trees, always requires evil ethics) removes the random chance aspect of flipping. Any eligible planet will flip on the first turn available (applies only to those flipping to whoever owns the structure, everyone else's flipping is uneffected). In prior version of the game, this was broken and instead of doing this the MCC gave a 100% economy bonus and actually inhibited flipping. If you see reference to "the MCC bug", that's what is being referenced. IIRC you don't have to worry about that, it was fixed before the Ultimate Edition came out.
I'm sure I've missed something, so if you have more questions just keep them coming.
