I believe it helps your game because you are practicing visualization skills. As these skills are practiced, your ability to visualize things in general increases, thus your ability see ahead in a game gets better.
How do square/color excercises help?

I'd say it helps because you know the square. I can't tell you the number of times I've watched a deathmatch, USCL game or some other commentated game hosted by chess.com and got COMPLETELY LOST when strong players start rattling off moves. They'll be like 4 moves in and I'm still looking for the 2nd move. I remember watching a deathmatch on here once with Sam Shankland as the co-host and he started going into moves and he just kept going. I was like forget it I'm making a sandwich!

You can also try this.
Reassign numbers 1-8 to the LETTERS a-h.
Then, add the numeric value of the file (letter) to the numeric value of the rank.
Even = dark square, odd = light square.
So:
e(5)+4 = 9, light
c(3)+5 = 8, dark
f(6)+6 = 12, dark
It's sorta my OCD.
You know those games or flashcards that drill your knowledge of what color each particular square is. People say that these exercises helps in "board vision" and thus helps in tactical vision, but how does it exactly?
As in, how does knowing that e4 is a light square and c5 is a dark square and f6 is a dark square help your game in practical terms?