Camus does not explicitly reveal his existential philosophy through his narration or his character's quotes.  He sets up the story to be an existential predicament.

First, look for quotes of alienation and aloneness.  The rocky terrain.  The barren landscape.  Daru's solitary schoolhouse.  The gendarme and the prisoner out in the snow.

Second, look at the choices that each character is given, and how each character avoids choice.  The gendarme hands over the prisoner to a civilian.  Daru tries to set the prisoner free.  The prisoner could have led a life of freedom, but chose not to.  These are existential predicaments.

Camus say that there are two choices: freedom and death.  The prisoner had a chance to join the nomads (freedom); instead, he walked to the prison (death).  What's worse, Daru made no choice at all.

How does "The Guest" by Albert Camus relate to existentialism?

While Camus himself did not identify as an existentialist, his works have had a significant impact on existentialism, and there are many elements of this philosophy to be found within "The Guest." In order to understand the importance of existentialism to this short story, it is necessary to dissect the major elements of existentialism and discuss how they present themselves throughout Camus's work.

Essential Nature

According to the tenets of existentialism, there is no essential nature in the world. This means that nature is entirely constructed of the choices each individual makes rather than the result of some inherent order in the universe. Another way to explain this concept is that nothing inherently has meaning other than the meaning and value the individual ascribes to it. Daru laments how little he and his unwanted guest matter to the harsh landscape of the desert, and yet neither of them would have any significant meaning outside of its barren context. While the prisoner seems to have the choice between freedom and imprisonment, fleeing would mean giving up all the meaning of the life he has created for himself.

Alienation

Both Daru and the prisoner are alienated from society by their own design. Alienation is a major theme throughout existentialism. Daru is alienated because he holds no allegiance to his country or to the society around him. The prisoner became alienated from society when he chose to commit a criminal act. Both men are alike in their alienation and the story's desolate setting reflects this.

Freedom

Freedom in existentialism is typically an illusory concept. The individual has the freedom to make his own decisions, but that comes with the burden of accepting the consequences of those decisions, which prevents him from being entirely free. The more freedom the individual obtains, the greater his isolation from society becomes. Daru illustrates this concept in his initial refusal to help the police. By refusing to take sides in the conflict between the French and the Arabs, he is cementing his own isolation from society. Existentialism leads to the realization that all restriction is self-imposed and, as the prisoner chooses to turn himself in at the end of the story, all individuals choose their own metaphorical prisons.

Choice and Consequence

Choice is a theme that runs throughout existentialism and the narrative of "The Guest," and it is related to all the other existentialist elements in the story. Daru makes an unusual choice to refuse to follow the orders of authority over his own morality, even though it will ultimately cost his life. Camus and the existentialists viewed morality as a construct of personal choice, and this decision marks the triumph of personal choice over obedience to authority. This choice comes with consequences, including Daru's almost inevitable execution at the hands of the Arabs who blame him for the prisoner's self-imposed isolation. This consequence can be seen as an existentialist metaphor for what can happen when rational choices are made in an irrational society.

What are examples of existentialism in "The Guest" by Albert Camus?

As another educator has mentioned, alienation is a common theme of existentialism, and one that plays a large part in “The Guest.”  Daru is completely isolated in his empty schoolhouse on the desert plateau, and by refusing to take the gendarme’s prisoner to Tinguit he isolates himself from Balducci, a previously friendly acquaintance.  His “guest” has isolated himself from society by his criminal act, for which, at the end of the story, he condemns himself when his village would willingly take him back.  This final point is important:  both Daru and the Arab willingly alienate themselves, and are dependent, as individuals, on the reality of this isolation.  As Daru notes, “No one in this desert, neither he nor his guest, mattered.  And yet, outside this desert neither of them, Daru knew, could have really lived.”  This is indicative of a key aspect of existential thought, Sartre’s idea that “existence precedes essence.”  We live, we breathe, we are, and yet this existence is wholly absurd and meaningless – none of it matters.  All the meaning we can find for our own existence is created by ourselves. 

By refusing to obey Balducci’s orders Daru is also exhibiting the existential notions of authenticity and freedom – that is, the use of one’s own free will to make one’s own decisions, based on the premise that life is uncontrolled by any divine set of rules, and that all rules are simply abstract fabrications of society, of the “crowd,” and by accepting the burden of responsibility for one’s own decisions, the individual is further isolating himself from this society.  By disobeying the local police, Daru is being true to his own beliefs and acting accordingly, a trait which is emphasized through addressing the Arab as his “guest,” rather than the prisoner or the killer or something equally damning.  And by giving his guest the same option – freedom or self-condemnation – he is acknowledging the Arab’s ability to do the same; to judge himself and react according to his own responsibilities.  We are all our own prisoners, and it is from ourselves that we create meaning and grace, hatred and punishment.  All that occurs in the world is happenstance, is neutral, and we much choose how to act on these occurrences and imbue them ourselves with any degree of importance.  And yet the fact remains, beyond and despite our efforts to do what we deem is right (according to our own analysis), that the world is indifferent to our struggles.  As Daru notes, when he sees the threat scrawled across the classroom blackboard upon his return from offering his guest a choice, “In this vast landscape he had loved so much, he was alone.”

[It should also be noted that although Camus’s works can (and should) be examined from an existential perspective, he did not identify himself as an existentialist philosopher].

In "The Guest," Daru finds himself in a situation that his moral compass is unable to deal with. By letting the Arab go free, he will run afoul of French colonial authorities, and by turning him in, he will anger the Arab's people. He wants to let him go free, but ultimately the Arab chooses to go to prison, which does in fact anger the people in his village, who blame Daru. Daru is stuck in a situation where his individual ethics conflict with the society where he lives, and by trying to take a middle course (by inviting the Arab to take a path toward freedom or prison) he is still unable to absolve himself of responsibility for events that are essentially beyond his control. The alienation and the notion that people sometimes have very little control over their lives are central themes common to existentialism.