Besides being the first true booster for art-related Great People, the Amphitheater also provides the first slots for Great Works in your empire. It won't do you any good to attract a Great Writer (if you manage it somehow, without this building) and then have nowhere to put their writings!
When bread and circuses didn’t keep the masses happy, there was always the theater to distract them. An amphitheater is an outdoor arena (modern usage of the term is rather lax) that serves as a venue for performances, sporting events and public speeches. Although there had certainly been public spectacles in various places before, these tended to be staged ad hoc. But when the Greeks began writing plays around 532 BC, they built small, open-air structures (called theatrons) in which to perform these; the Romans, always looking to build bigger and better, created the amphitheater. The Romans built them all over their conquered lands, for every frontier outpost wanted to be civilized. The practice spread outward from the empire to other places, for amphitheaters were great places to stage plays, gladiatorial and Olympic Games, religious spectacles, executions, political debates, concerts, and all manner of other entertainments.