Re: work/play/go through/go over
I'm a lifelong (amateur) chess player, and I don't believe the expressions you've quoted are unique to chess.
Echoing emsr2d2 and Raymott, I take both of them to mean the same thing: The author is going to provide you with a series of chess moves, and he expects you to copy them one by one, presumably on a chessboard or a computer, paying attention to the situations and complications that unfold.
The author is giving you a chess lesson, and to put it in longer context, he's saying: "If you want to become a better chess player, study the moves of this game from start to finish. Notice how [Player A] gets the better of [Player B] by [attacking his pawn structure, etc.]."
I wouldn't spend a great deal of time analyzing it -- the author is simply trying to improve your chess skills by having you copy the moves of a previous game that he finds instructive.