"Fini" similarly like "komenci" is a transitive verb. So for example: "Mi finis la ludon" means "I finished the game". There are a subject: "Mi", verb: "finis" and the object of the verb: "la ludon". "Finiĝi" is an intrasitive verb so it doesn't have an object. "La ludo finiĝis" means that the game finished but it doesn't mean that the game was finished by someone or that the game finished something.
I guess it's because you do not play a play. You may act out a play. Likewise you generally do not game a game but just play it. The words in English are different. But languages like Esperanto or German use the same word stem:
a game = ludo/ein Spiel
to play = ludas/spielen