Why is the old man at the bridge?
The old man has just crossed the bridge and has sat down beside the road because he is too tired to go any farther. He is one of the many civilians fleeing before the advancing forces of Generalissimo Francisco Franco, the Fascist leader who eventually became supreme ruler of Spain. The Fascists, or Nationalists, were killing peasants and workers as a means of spreading terror and also under the assumption that these impoverished people must be supporters of the lawfully elected Loyalist government. There were terrible atrocities committed by the Fascists during the war.
The narrator is standing there because he is watching for the approach of the Nationalist tanks, trucks, and soldiers. He is evidently an American and one of the many foreigners who volunteered to help the Loyalist cause against the fascist rebels. Hemingway wrote at length about the Spanish Civil War in his best novel, For Whom the Bell Tolls. It features an American volunteer assigned to blow up a bridge to stall the advance of Fascist tanks. The dialogue in "Old Man at the Bridge" is in English, but it is to be understood from the simplicity of the vocabulary and some of the syntax that the narrator and the old man are speaking in demotic Spanish. Hemingway did this in For Whom the Bell Tolls. He was very good at writing dialogue and used dialogue to characterize the speakers as well as to provide exposition.
The narrator is primarily concerned about the advancing army, but he is also concerned about the old man.
"This is not a good place to stop," I said. "If you can make it, there are trucks up the road where it forks for Tortosa."
When the narrator again advises him more urgently to get up and go on,
"Thank you," he said and got to his feet, swayed from side to side and then sat down backwards in the dust.
It is pretty obvious that this old man is going to get killed when the Fascists cross the bridge. He is too old and too worn out to flee any farther, and he doesn't really have anything left to live for. He might be said to symbolize the Liberal government and its Loyalist supporters, gradually being overwhelmed by the Fascists who were backed by Hitler's Germany and Mussolini's Italy.
This little story, or vignette, or slice-of-life, is intended to represent the big picture of the war. Hemingway was a foreign correspondent and was accustomed to sending brief dispatches because of time and communication constraints. Foreign correspondents were fond of finding little scenes that symbolized great historical events. Hemingway wrote this story as a dispatch and then decided to publish it as a short story instead. Many of Hemingway's dispatches are collected in an interesting book titled Byline Ernest Hemingway.