How do the allusions to existentialism and nihilism help the reader understand the theme of "A Clean, Well-Lighted Place"?
Existentialism says, in brief, that life is meaninglessness but that each individual has the personal necessity to try to sort out a value system or moral system by which to live in a thus self-ordered world so as to create meaningfulness and to prevent falling into despair.
Nihilism says, in brief, that any attempt to create order is futile and despair is inevitable. In other words, nihilism takes existentialism one step further while scorning its effort at creating order in a meaningless and orderless world and, in so doing, nihilism declares there is no possibility of ultimately imposing order or meaning because in the end, all is death and destruction. Nihilism then posits that if this is so, all efforts against meaninglessness are meaningless.
The old waiter expresses nihilism and existentialism in a combined existential nihilism in his opening remarks about the old man:
[Nihilism]: "He was in despair."
"How do you know it was nothing?"
[Existentialism]: "He has plenty of money."
Existentialism would suggest that, given ample money, the old man might successfully create some meaningful or moral order with which to offset despair. Nihilism says off-setting despair is impossible because all ends in meaningless death, which the old man is approaching and doing so alone (his wife has died already).
If you wish to think of these existential and nihilistic dialogic remarks as "allusions," then these allusions to existentialism and nihilism don't simply "help the reader understand the theme," they represent the theme that they systematically build: thus the reader is "helped" by hearing existential nihilism expressed in common terms by ordinary characters. A central theme of this story is an existential one that suggests that while nihilistic meaninglessness is strongly felt, there may indeed be ways to offset the pull of nihilism and the strangle-hold of despair.
One suggestion to defeat nihilism Hemingway makes in the story is that of the security of a clean, well-lighted place. That a place is clean and well-lighted suggests an innate belief in meaningfulness: if all were meaninglessness, there would be none to clean and make a place well-lighted. Another suggestion is the immediacy of intimacy represented by the soldier, risking arrest, walking quickly down the street with a girl hurrying alongside him:
The street light shone on the brass number on his collar. The girl wore no head covering and hurried beside him.
"The guard will pick him up," one waiter said.
"What does it matter if he gets what he's after?"
Another suggestion is that of having "plenty of money ... a wife ... youth, confidence, and a job." These things provide the rewards of emotion, connection, accomplishment, satisfaction and opportunity. These rewards in turn provide a sense of order and meaning in a world that is devoid of order or meaning. This sense of order fends off feelings of despair by providing something to look forward to and desire and feel pride in: "I am of those who like to stay late at the cafe, ... because there may be some one who needs the cafe."
Hemingway makes a further suggestion through the older waiter's behavior and final thoughts as he lay in bed: "finally, with daylight, he would go to sleep. After all, he said to himself, it's probably only insomnia. Many must have it." Apply an analysis to the waiter's speech about "nada" and to his final comments to see what you can find about Hemingways' final suggestion(s).