Chess notation explained
Your example seems to be an older or more classic notation where the file/column is indicated by the name of the piece that starts on that file, rather than a letter. So for something like "king to rook three", "king" is the piece that moves, "rook" is the file (I assume closest to the piece that moves, since there are two "rook" files on the far left and the far right of the board) and "three" is the rank indicating the number of rows from the white side of the board.
Actually if I recall correctly, if there is any confusion about which file is meant by "rook", "knight", or "bishop", you can specify with something like "queen's bishop four" meaning the file of the bishop that starts on the queen's side of the board; e.g. "knight to king's rook four".
Still, you can see how this might still lead to some confusion when there is more than one piece which can move to a particular spot, I'm sure there are either some way to specify, or perhaps skilled players know that only one move makes sense.
I also assume that Fischer is alternating player's moves, so that "King to rook three" is one player's move, "knight to king six" is the other player's move, and so on.
Finally notations for moves like "pawn takes queen" are simplified and assume that skilled players already know which pawn is meant, since only one pawn should be in the proper position.