The overhearing of the conversation between father and doctor is what triggers the boy's concern over dying.  It becomes clear that the boy perceives the conversation between the father and the doctor in a fatal light.  When he hears the disclosure of a 102 degree temperature, the boy confuses this with his own thoughts about the capacity for human temperature.  The immediacy of death becomes real at the convergence with the boy's own misunderstanding of the difference between Fahrenheit and Celsius readings of temperature as well as his father leaving him.  Driven by both realities, the boy becomes concerned over his death.  

The boy believes he is going to die and that the "wait" is for death.  The boy becomes overcome with a reality over which he has no control, but something that obsesses him nevertheless.  It is in this where the ending of  how "he cried very easily at little things that were of no importance" becomes relevant.  The dwelling on death over misunderstanding and elements that fundamentally represent "no importance" is what causes the boy to become concerned over dying from the very exposition of the story.