In order, not merely to enjoy the Hemingway story, but to understand it, the reader must do two things.  Firstly it is necessary to observe the two waiters very closely.  We need to understand their characters and motivations.  Ultimately, we need to understand what they represent within the story.  We need to know what types they are, in order to be able to attribute language to them; in order to achieve the second  thing, to know, on a very mechanical level,  who is speaking to whom.  The reason that it is at all difficult is that the waiters, echoing  real conversation, don't alternate speech, like a ball bouncing back and forth across a net, but hesitate, speak again.  Interject, just as we all do.

Earlier, we noted how Hemingway leaves out the speech tags of the dialogue, creating a spare, curt style.  The side effect of the removed apparatus is a dialogue filled with silences and spaces.  Thus, while the narrative drives us towards comprehension, it simultaneously reminds us of the dislocation of these characters, their relative isolation and loneliness -- literally the silences surrounding them. 

We cannot read the story without striving to understand it.  And having come to terms with the two waiters and what they individually represent, we are more able to understand the older waiter's affinity with the old man.  What is more, his sympathy and feeling, becomes ours.  We come closer to understanding the feeling of despair that is expressed in the nihilistic litany "Our nada who art in nada...."  It is a feeling of an individual and perhaps of the post war generation as a whole, engaged in the existential rediscovery of themselves, and finding only that restless, rootless insomniac wandering through the night in search of a "clean well-lighted place."