Three times in the poem, Yeats ends a stanza with the phrase, "A terrible beauty is born." He even ends the poem with it, which should set off our spidey sense and tell us that it's probably an important phrase.
Whenever he talks about terrible beauty, Yeats seems to be trying to bring together the different (even contradictory) emotions he feels when he thinks about the Irish Uprising of Easter, 1916. On the one hand, the Uprising is beautiful because it'll go down in history as a great fight for Irish freedom. On the other hand, a whole bunch of people died. The phrase "terrible beauty" seems to be Yeats's way of saying that history's most celebrated moments are usually moments of death. A little dark, but kind of true.
Lines 15-16: So far, Yeats has been going on about how he doesn't really care about his run-ins with the common folk of Dublin. But in lines 15 and 16, he says that everything is suddenly "changed, changed utterly" and that "A terrible beauty is born." People who understand the reference to the Easter Uprising in the poem's title no doubt understand how everyday life would have changed when the fighting started. But at this point, we're still not sure what Yeats finds particularly beautiful about this. We'll have to wait for our answer.
Line 40: After talking about a bunch of people who died in the Easter Uprising (or were executed later), Yeats repeats the phrase, "A terrible beauty is born." Again, it's kind of hard to tell what's so beautiful about all these people dying. But by this point, Yeats is sort of getting at the idea that these people are going to be remembered for the brave things they did, and there might be something beautiful in that.
Lines 79-80: By the end of this poem, we're starting to get a clear picture of what Yeats has meant when he's said, "A terrible beauty is born" throughout this poem. He's saying that in the future, the people of Ireland will remember those who fought for Irish freedom. And on top of that, Yeats is reminding us that all of the big historical moments we tend to consider meaningful or "beautiful" tend to be moments when a ton of people died (American Revolution, anyone?).