While the other answers given are correct to a point, they are only correct to a point.
Golf can be scored in one of two way: stroke play (which is currently the most popular method, counting the cumulative number of strokes taken by each player over the entire course), or match play, which was once more common in tournaments, and is still the most popular form of one-on-one competition involving wagers. In match play, a player wins by winning the largest number of holes played, and each hole is scored by strokes individually. Anyone whose golf game is not strong enough to play "scratch" (likely to play the course at or near par) is given a "handicap" of one or more strokes, which they are allowed to deduct from their total strokes on a given hole (and on most courses, the order of holes on which handicaps are to be applied is usually provided - you can't simply use your handicap strokes where they would give you the best advantage). In this case, the handicap being offered is one stroke on each hole rather than, say, eighteen strokes over the entire course, so the stronger golfer is betting that he can beat the weaker by at least one stroke on more than half of the holes on the course.