The theme of "The Old Man at the Bridge" is that war not only threatens to destroy the lives of soldiers on the battlefield, but all living beings who happen to be in its path. This anti-war story focuses on what we today would call war's "collateral damage."
The old man is one example of collateral damage. He has been living his ordinary, peaceful, compassionate life in a rural village in Spain. He has no idea what the fighting is all about. He hates nobody, has no politics, and simply wants to do what he has always done, which is tend his animals, but he is forced from his home. He has nowhere to go, nobody to stay with, and no desire to start over, yet he has to leave.
Through his concern for the fate of the animals he has left behind, we learn about even more of war's collateral damage: he has let his cats run free and has left his bird cages open, but what of his farm stock? He worries about the bombs dropping from planes and an enemy army coming through. What will happen to his poor animals? Like him, these are creatures without any politics or hate whose lives are threatened and upended.
Hemingway wants to make sure that we don't forget about all the vulnerable creatures, from animals to elderly people, who suffer from wars. He is saying that these lives count, too, and are part of war's toll.