Ballads tell a story in rhyme, and in "The Ballad of Father Gilligan," Yeats's speaker relates the story of how on one particular evening, God showed mercy on both a tired old priest and a man at the moment of his death. The theme is God's omnipresence and grace in the lives of the faithful.

The first stanza implies that Father Gilligan's congregation is dwindling; half are either dead or confined to bed. Father Gilligan is an old man whose religious duties are physically tiring, and he struggles with the limits of his aging body when duty calls in stanza two.

In his infinite mercy, God allows Father Gilligan to fall asleep and rest through the night while his parishioner is dying. He sends an angel to the dying man's house in Father Gilligan's place, and the man's widow identifies it as Father Gilligan when she speaks to him in the tenth stanza.

The last two lines of the poem observe the theme: though God has "planets in His care," he attends to the physical and spiritual needs of all his people—including an exhausted priest, who humbly considers himself "the least of things."

I would say that the theme of the poem is the idea that God cares about the least members of his creation.  In the Bible it talks about how God knows if even one sparrow dies.  This poem's theme is the same as the idea that is in that quote.

In this poem, Father Gilligan is worn out emotionally and physically.  So when he tries to pray for the "poor man" who sent for him, he falls asleep.

When he wakes up the next morning very early, he goes to the poor man's house and finds that an angel in his own form had already been there.  The angel helped the man die happy.

Gilligan says this shows that God cares even for him and the poor man and that is the theme of the poem.