Cabe says: Old Cotter speaks in unfinished sentences, so that the boy, who is deaf to his innuendoes, thinks him a "tiresome old fool."  But it isn't hard to locate the source of his ellipses.  We presume that he didn't talk in unfinished sentences when he used to talk to the boy about "faints and worms."  But now, because he has something that approaches an opinion on Fr. Flynn's death, he withholds his words when the boy can  hear. 

What is it that is keeping him from speaking to the end of his thought?  For one, he doesn't think the boy ought to know or hear such thoughts.  And so he does not speak what should not be heard.  Beyond that, though, he seems to have trouble confronting them himself.  He says there was "something queer, something uncanny" about the Priest.  The OED shows "uncanny" to mean untrustworthy, unreliable.

We could explore the word Old Cotter uses, if you wish...