There may be a good reason you're having trouble figuring out the tragic nature of Synge's The Playboy of the Western World:  it isn't a tragedy.

The play deals with conformity and rebellion, with myth making, and with mob mentality.  It also presents a not-so-complimentary portrait of the Irish.  But it isn't a tragedy.

Nobody dies, first of all.  Second, fate doesn't lead to anyone's downfall.  No life-changing battle is fought--figuratively or literally--by the protagonist.  Nothing is cleansed away by a purge at the close of the work.  In short, it's not a tragedy.

If one had to think in these terms, Playboy may be an anti-tragedy.  The protagonist tries to kill his father, but fails; he is at first made into a hero for his rebellion, but later almost killed when the fickle crowd changes its mind; the hero is not a hero; the crowd's penchant for creating myths is exposed as faulty; and the play is more comic than tragic.  It is almost mock-tragic, in the same sense as something mock-heroic or mock-epic.

The villagers are bound to tradition, prone to create heroes where there are none, and violent.  Their early admiration for Christy's violent act is certainly a comment on the Irish mentality.  But it isn't a tragedy.  For that, you would need to look at Synge's "Riders to the Sea."