In a nutshell, the moral of any story is the lesson that the author wants you to take away from it. Since "The Moose and the Sparrow" ends with a bully dead in a ravine, I would argue that the lesson being taught here is to be kind to those around you, even if you don't like them or if they are from a different walk of life.
This message is driven home by the discovery made by our narrator right at the end of the story. The fact that there are marks in the trees on the bridge from which Moose Maddon fell indicates that someone may have set up a trap or a tripping hazard. In turn, it would then seem that drunkenness was not the cause of Moose Maddon's death. Based on the evidence, it would seem that Moose Maddon's ongoing unkindness to Cecil was the reason for his untimely demise. Since Cecil had been making a watchstrap from wire for Mr. Anderson at the time, we know that he had plenty of wire and a good idea of how to use it effectively.
Being unsure of what happened, and possibly understanding why Cecil may have chosen to do such a thing, Mr. Anderson keeps his discovery to himself. This means that Cecil is not punished for Moose Maddon's murder, which in turn reinforces the message that bullies will not meet a good end.