A few things.
Pieces can get to advanced or important attacking squares and can't be removed easily.
You can target weak pawns on a certain colour.
Restrict the movement of opponents pieces based on your control of a certain colour.
You can blockade passed pawns without your piece being dislodged.
It can be critical around the king e.g. if your fianchetto'd bishop on your kingside gets exhanged off pieces can get to f3/h3 to attack the white king and vice versa.
To a certain extent just being able to add another piece to an attack on either light or dark squares may be hard to defend.
Not any rules I've read, just trying to think logically.
Obviously, I'm guessing it just means that you control either the dark or light squares, but my deeper question is what does that accomplish?
I understand controlling the center or maybe controlling a single file, but what does controlling either the dark or light squares accomplish and allow for (i.e., what can you do with it)? I hear this a lot of "expert" chess commentary/analysis, but it's not obvious to me how to use it to exploit your opponent.
Thanks!