Thomas Kincaid, a medical student at the University of Edinburgh in the 1680s, kept a diary. In it, on January 21, 1687, he wrote the first known reference to handicapping, discussing options of betting. "At golf, " he wrote, "whether it is better to give a man two holes of three, laying equal strokes, or to lay three strokes to his one and play equal for so much every hole." Kincaid was comparing types of betting; was it better, he asked, to give a player a two-hole start every three holes and play with no strokes, or play even, paying three-to-one odds per hole? Captain Elphinston challenges Mr. Allan next Saturday best of three rounds, half a crown (currency) a hole, that he beats Mr. Allan with the Club against his throwing and gives him half one. No running at the throw! That match was halved.
Mid-19th Century professional golfers like Allan Robertson often derived a significant portion of their incomes through wagering on golf. The concept of giving strokes allowed Robertson to set up matches with golfers who weren't at his level.
Match play is the older form of golf, and the only one mentioned in the rules until ca. 1807. As shown by the second paragraph, handicaps were about money, and in Robertson's case it was the only way to get a match with anyone other than a professional.
'Medal' play was just that, stroke play for a medal and it had monetary value, and in at least one instance that I came across the winner of the medal was paid half a crown to wear it during certain club functions.
Once again, it's money.