Who was talking to the old man?

The narrator of this brief story does not identify himself, but there are indications that he is one of the many foreigners who volunteered to help the Republican (Loyalist) cause during the Spanish Civil War, which broke out in 1937 and ended with the victory of the Fascist forces led by General Francisco Franco, aided and abetted by fascist Germany and fascist Italy in what is generally considered a prelude or rehearsal for World War II. Ernest Hemingway dramatized the Spanish Civil War in For Whom the Bell Tolls (1940). In that excellent novel the hero Robert Jordan is an American schoolteacher who is fighting alongside Loyalist Spanish guerillas, and it would seem that the narrator of "Old Man at the Bridge" is someone like Robert Jordan. The narrator tells us:

It was my business to cross the bridge, explore the bridgehead beyond and find out to what point the enemy had advanced. I did this and returned over the bridge. There were not so many cars now and very few people on foot, but the old man was still there.

Although the dialogue is in English, the reader is to understand that the two men are really speaking Spanish. The old peasant certainly would not know English. The narrator's Spanish/English is somewhat constrained because he does not know the language well. We can tell this because he speaks to the old man in short sentences using a limited vocabulary, although his explication to the reader is in proper English, as in the long sentence:

It was my business to cross the bridge, explore the bridgehead beyond and find out to what point the enemy had advanced.

He would not have tried to say this to the old man in Spanish/English because he wouldn't have known how to say it in Spanish, and the old man would never have understood anyway. An example of the narrator's stilted Spanish is the following:

"What politics have you?"

This is the way the question would be translated from Spanish into English if it were a literal translation. Hemingway does this sort of thing with dialogue all the way through For Whom the Bell Tolls, and he also does it in his long story "The Old Man and the Sea" (1952). He does it very well. The reader is made to feel that he is hearing people speaking in Spanish when the dialogue is actually in English.

So the best answer is that the narrator is an American volunteer working for the Loyalist cause and trying to carry on a conversation with a aged native Spanish speaker while his attention is mainly focused on the area on the other side of the bridge, where he expects to see the Fascist army appear momentarily. The narrator has obviously been in Spain for some time and has seen fighting, air raids, artillery shelling, fleeing civilians, countless corpses, and other grim aspects of the Spanish Civil War.

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.