Hemingway is presenting in "A Clean, Well-Lighted Place" a picture of existential nihilism and arguing against nihilism while acknowledging the truth of existentialism. The older waiter is the character who draws the picture for us, as a result, Hemingway has given him depth of understanding of life's experience with existentialist and nihilistic feelings.
Existentialism essentially says that life is meaningless and without order and each person has the necessity to create order and meaning on their own for their own lives in order to fend off ultimate hopelessness and despair that comes from being a thinking being in a senseless world. Nihilism goes further and essentially says that the end of everything is death and destruction therefore every attempt to create order and meaning is itself meaningless.
The older waiter understands the old man's need for the clean, well-lighted cafe because he (1) understands the old man's despair. The old man no longer has order and meaning in his life because his wife has already died, and we are to suppose that it was union with her that gave life order and meaning for the old man. The waiter understands that (2) now the only order the man can find is the order of externalities and the only meaning, that of some sort of activity and human exchange. These substitutes for meaningfulness don't work well for the old man as we at the start of the story.
The waiter understands these things so well because, as we learn in the latter portion of the story, he is having his own existential battle against nihilism. He feels life is meaningless and "nada." He feels the only order in life is that imposed by externalities, like cleanliness and good lighting in cafes. He finds his only meaning in (a) the human exchange that is possible because of keeping a clean well-lighted place open for others who are in need and in (b) the human exchange of thinking that others also suffer his form of "insomnia," thus he doesn't suffer alone.
The cafe represents to both of them an accessible option to fight against nihilism, against that final hopeless despair of non-meaning for a reasoning being: we can reason, yet we can find no meaning through our reasoning while we watch as all comes to death and destruction.
"It is not only a question of youth and confidence although those things are very beautiful. Each night I am reluctant to close up because there may be some one who needs the cafe."