"Fantasy" is a strong word to use in relation to a play noted for its realism (not originally favorably noted, either). One definition of "fantasy" is: "imagination unrestricted by reality" (Collins English Dictionary). While the things that happen "near a village, on a wild coast of Mayo" may be "the wildest sayings and ideas," to quote Synge, he is noted for attempting to attain an untainted and real image of the Irish on the Aran Islands. Having said this, it can be agreed taht the villagers did have one or two wee fantasies a-goin' on in their brains.
The theme of fantasy versus reality is established when a stranger drags himself into a small remote village and asks if the police come there often. Sober-minded reality goes from the villagers' minds, collected as they are in the local inn, as fantasy takes hold and they start guessing at what criminal act he may have committed so that he is avoiding the police. Their fancies run wild until they drive him to confess that he just killed his father and ran for it. Rather than realistically show him the door and the quickest way out of town, they continue to fantasize about him. Eventually they lionize him and give him the title Playboy of the Western World as his reputation with them overrides his actual life and he becomes the champion of every local sports game they play.
Reality intrudes in the form of his father Mahon, who survived Christy's attack, and has come to search him out and, while there, set the fantasies straight. As a result of Mahon's physical attack upon Christy, during which the villagers discover he did not kill his father--only tried and failed!--the villagers turn on him.
PEGEEN — [glaring at Christy.] — And it's lies you told, letting on you had him slitted, and you nothing at all.
The end result is that reality forces Christy back to his father's notice and ends his engagement with Pegeen. Reality gives him the opening to march away from the village and Mahon's control as a man whose fantasy has become reality:
CHRISTY. Ten thousand blessings upon all that's here, for you've turned me a likely gaffer in the end of all, the way I'll go romancing through a romping lifetime from this hour to the dawning of the judgment day.