This is a very good question. It does seem as though the narrator is spending a lot of time talking with the old man under the circumstances. The narrator is expecting to see the enemy forces arriving on the other side of the bridge, and he must be thinking about his own safety, because he will have to leave in a hurry when they appear.
I was watching the bridge and the African looking country of the Ebro Delta and wondering how long now it would be before we would see the enemy, and listening all the while for the first noises that would signal that ever mysterious event called contact, and the old man still sat there.
Hemingway was a foreign correspondent and wrote a number of dispatches to American newspapers for four decades. They are collected in a book titled By-Line Ernest Hemingway (1967) and make interesting reading. It was typical of foreign correspondents to pick out little scenes that were symbolic of big events, and "The Old Man at the Bridge" sounds very much like a dispatch from a war correspondent. I believe Hemingway wanted the story to sound like such a wartime dispatch, written in a hurry under dangerous conditions.
The main reason the narrator seems to be talking to the old man at such length is that he is concerned about his safety, but he wants to establish some degree of mutual trust before advising him to flee. The old man should obviously not be sitting there. No doubt the fascist soldiers would shoot him if he was still there when they crossed the bridge. He is a peasant, and the Nationalists under Francisco Franco were taking severe reprisals against the peasantry under the assumption that all peasants were Loyalists (or Republicans). Hemingway describes all of this in detail in what many consider his best novel, For Whom the Bell Tolls (1940). The most important part of the conversation between the two men comes after the narrator has found out a little about the old man, including his politics.
"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."
"I will wait a while," he said, "and then I will go."
The narrator may be making small talk with the old man, but he is watching and listening for the approaching enemy at the same time. The approach of the unseen enemy seems to haunt the scene. It is what provides the drama for this narrative. We can feel and almost see them--some marching on the dusty road and some packed into open-bed trucks. There would also be a few old-fashioned rumbling tanks.