"The Fly" is the story of a man, known only as "the boss," who has been utterly devastated by the death of his son. The boss's son was killed in World War I. When he first found out about his son's death, via a telegram, the boss was left "a broken man, with his life in ruins." In other words, the boss was, understandably, completely devastated and felt as if his world had come crashing down around him.
"The Fly" is set six years after the boss's son's death, and the boss feels guilty because he can no longer shed a tear when he thinks of his son. When he first learned of his son's death, the boss would regularly be "overcome by such grief that nothing short of a violent fit of weeping could relieve him." Now, however, six years later, he is unable to feel the same degree of grief. He feels ashamed that "no tears [come]" when he looks at a photograph of his dead son. The implication is that the boss has been drained of all emotion. He is no longer the person he used to be. This is why he is only known as "the boss" in the story. This title suggests that there is no sense of self left. He is defined now only as the boss of a company. He is no longer a father and thus no longer the person he used to be, that person having been defined in large part by his role as a father.
The idea that the boss has been reduced to a title since the death of his son is emphasized at the end of the text. At the end of the story, the boss decides to torture a fly caught in an ink pot. He tortures this fly in order to try and feel something again. The death of his son has left him completely numb.