Firstly, in golf, 'strokes' are how you score the game:
What is a stroke
tl;dr: every time you swing the club at the ball, it's a 'stroke'. When the game is over, the player with the _least_ strokes wins.
In this instance, 'a stroke a hole' means the 'better' player is handicapping himself to give his opposition a better chance.
Stroke hole, handicap hole or handicap-stroke hole
So, over 18 holes, the better player is giving the opposition 18 'free' strokes.