Advertisers strategically place products they would like to sell in game shows as prize winnings or as prize money "sponsors." Without paying for advertising time, this allows them to show their product favorably to all the potential customers who watch the show and give the advertising money to the show, which can use some of it for prize money.
Manufacturers of products also work closely with game shows. While the game show provides product advertising or credit to the manufacturer for sponsoring the prize, the manufacturer throws in some money for the winner.
Commercial advertising pays for all network television shows. Game shows are no different: The revenue generated from commercial advertising pays for the show. Because game shows are relatively cheap to produce, this allows the producers of the show to set aside prize money from the commercial advertising, too.