A caddy is a person who carries a golfer’s golf clubs and, sometimes, assists the golfer in choosing which club to use. “To caddy” is a verb that means to serve as someone’s caddy. That’s what it means here: he served as the woman’s caddy and got paid a little bit for doing the job.

A 'caddy', and the related verb 'to caddy', are golfing terms. In golf, the players use a variety of clubs (the stick) to hit the ball, and the person who carries those clubs around for them is called a 'caddie', and what he is doing is called 'caddying'. Thus 'I used to caddy' means 'I used to carry around golf clubs during golf games, for money'. By the way a 'club' means both the stick that they use to hit the ball, and the place where they play.