Here's the context of the line you ask about from "The Ballad of Father Gilligan," by Yeats.

The old priest Peter Gilligan
Was weary night and day;
For half his flock were in their beds,
Or under green sods lay.

Once, while he nodded on a chair,
At the moth-hour of eve,
Another poor man sent for him,
And he began to grieve.

‘I have no rest, nor joy, nor peace,
For people die and die’;
And after cried he, ‘God forgive!
My body spake, not I!’

He knelt, and leaning on the chair
He prayed and fell asleep;
And the moth-hour went from the fields,
And stars began to peep.

I've emboldened the lines you ask about. 

Poor Father Gilligan is overworked and exhausted.  Members of his flock are dying faster than he can get to them to comfort them and deliver the Last Rites to them.  At the end of, apparently, another in a long series of long days, he is dozing off in a chair when he is summoned by still another dying man.  He reacts in a way that is really quite natural, but he sees it as a terrible sin. 

As soon as he realizes what he has said, he begins pleading for forgiveness.  He says that it was his tired and exhausted body that said what he said, not his mind.  His body is tired, but his mind didn't mean what he said.

In the poem, God seems to understand.  Even after the exhausted priest falls asleep while he is praying for forgiveness and misses the man's death, God sends an angel in the form of the priest to the dying man to administer the Last Rites.     

In the Bible, there is a verse that says something about the spirit being willing, but the flesh being weak.  This is what Father Gilligan is saying in this line.  He is saying that his spirit is willing, but his flesh is weak.

Father Gilligan is saying that he did not really mean to complain.  He is saying that it is just that his body is so tired that he wants rest.  He is saying that his spirit is willing to go out and do the work of a priest -- the work God wants him to do.