How is the story's plot advanced by the dialogue?
An interesting and subtle question. Because there is almost no plot per se in this story, you could possibly say, "Not at all," and just stop there. (There is little pursuit of a goal, and the story is so short that there's almost no sequence of events to provide a story line.)
However, if we define plot more broadly, the plot is advanced considerably by the dialogue. The dialogue tells us about the characters, and, more importantly for the plot, it provides the emotional momentum that carries the older waiter into that final explosion of despair the defines the story and gives it its meaning. The dialogue creates pressure, by the back and forth, and it introduces symbolic elements (the light, the darkness, age, fear), all of which are essential for that eruption of private despair.