Dunno how universal this is, but the Colosseum/Flavian Amphitheater had a massive underground section where supplies would be stockpiled for the games. The underground section is exposed to the open air today, so if you visit you can see it from above. As for getting the supplies down there, teamsters and dock workers have been among the largest urban professions since the dawn of civilization, so the answer is a lot of overworked slaves and proletarii.