Father Gilligan finds that he has slept through the night and failed in his duty to one of his flock that had sent for him the night before. Here's paragraphs six and seven from "The Ballad of Father Gilligan," by Yeats:
Upon the time of sparrow-chirp
When the moths came once more.
The old priest Peter Gilligan
Stood upright on the floor.
‘Mavrone, mavrone! the man has died
While I slept on the chair’;
He roused his horse out of its sleep,
And rode with little care.
The moths in Ireland come again in the morning, and the sparrows chirp. Father Gilligan wakes up, startled that he has slept through the night, and jumps up. He yells that he has missed the man's death, prepares his horse, and rides as fast as he can to the home of the man that sent for him the previous night, before he fell asleep while praying.
Of course, the Father arrives at the house, and discovers that God, in his mercy, has sent an angel in his place to administer the Last Rites to the dying man.
Right after he wakes up, he doesn't really find out much of anything. He gets very upset and worried because the man must have died while he was asleep (so I guess he has realized that it is morning).
It is only later in the poem that he finds out anything important. He finds out that, yes, the man has died. But someone who looks like him (Gilligan) has been to the man's home and has comforted him as he died.
Gilligan decides that the person who looked like him must have been an angel sent by God.
In the 7th stanza Father Gilligan wakes up from his sleep and realizes that he has fallen asleep even while he was kneeling down and praying - he was so tired and exhausted. He realizes that he has not attended to the dying request for the last communion of one of the villagers and saddles his horse and rushes off in a tearing hurry to administer the last communion to the dying villager and thus hoping to save his soul:
'Mavrone, mavrone! the man has died While I slept on the chair'; He roused his horse out of its sleep, And rode with little care.
Father Gilligan sees that the night has grown darker ( there are more stars) and as a result grows flustered that the dying man that he was supposed to visit may have died without the necessary absolution and comfort that priests are supposed to provide. He feels that he has let him down and not done his duty through succumbing to his weariness.