Is it true there is a website that pays you to play video games

No, that would be unlikely. While there are many sites that let you play for free, most gaming websites are for-profit enterprises, and they want you to buy something, which is how they make money. They may charge a membership fee or expect you to see advertisements from sponsors. Some sites might try to lure you by giving you a certain number of free plays, but then they would expect you to purchase more plays or tokens or whatever. So it is not likely that a site would pay you to play, unless you were one of their testers and were being paid by the developers to find potential problems with the game.