Starbuckone, the bookies' runner was needed years ago. Back then, you could only place cash bets on the racecourse or at the dog track. There were no betting shops where ordinary people could bet. If you were wealthy enough, you would have a credit account with a bookmaker; no cash needed, very convenient, phone betting and all, but not available to most.
To get round all this, bookies would have runners. These men would visit pubs , shops, workplaces etc and take people's cash bets to the bookie. They were operating illegally but they were the means for most people to get bets on.