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.