In the example you gave, it sounds like it's using the word "control" in the same sense that we're told to "control" the center, which I take to mean having an influence on that part of the board, which consists of four squares instead of one. I like to use my own term "forcelight" to describe the situation where a unit is throwing its line of force onto a square or unit, whether the target unit is his own or his opponent's, and whether or not the forcelighting unit can win any material there. That's a more accurate term, even if nobody else wants to use it.
In your example,
White's c4-pawn would forcelight the squares b5 and d5
White's d4-pawn would forcelight the squares c5 and e5.
Put together their combined influence and you get the list you specified: squares b5, c5, d5, and e5 are forcelit.
What good is that? Well, it's unlikely any of your opponent's pieces will want to land on any squares forcelit by pawns or by any units of lesser value than the unit landing there. Also, in cases where entire lines of access are involved, such as doubled rooks on a d-file, it is often the case that whoever can forcelight that file first (especially the opponent's Q1-square), can prevent *any* of the opponent's pieces from occupying that file for a while, sometimes permanently. Obviously that is a huge advantage, probably a winning advantage in most cases.
I'd read something when the book describes "fighting over control of a square" or "white has control over squares". What does this exactly mean? What good does having empty squares do for me? Is this related to prophylactics?
Example:
The white pawns c4 & d4 are hanging pawns. (There are black pawns b6 & e6)
Pros: Gives white control over the squares b5 c5 d5 e5 <---- What does this exactly entail?
Cons: They are weak since it's hard to move