Without a single shot fired or a drop of blood spilled in the story, Hemingway portrays where the true destruction of war lies in "Old Man at the Bridge". How does the story highlight the destruction?

The story conveys the ravages of war in a variety of ways. Often, the old man is frequently discussed regarding this type of question. He most certainly seems to be in a mental state of complete loss. He can't mentally move past the fact that he has been displaced by the war. Specifically, he mentions artillery strikes, and he is worried about what will happen to the animals. The soldier does convince the old man to get up and try to move, but his age and frailty prevent him from going anywhere. The man conveys a general sense of loss and hopelessness.
Beyond the old man, the story does a nice job of conveying the dismal effects of war through the use of color. Readers are told about the grays and blacks that are present in the faces of people and the surrounding area. Everything is covered in dust too. These are specific choices by the author. Whether there is a war or not doesn't effect the weather, yet the author tells readers that the overcast skies were so thick with gray that the planes couldn't even fly. The sky could have been blue with white clouds, but that weather and color palette would not help to convey the ominous overtones that go with war and coming battle.

The story is set during the Spanish Civil War, a particularly brutal and bitter conflict that dragged on for three years. And as the story opens, a huge convoy is making its way across a bridge in anticipation of the imminent arrival of enemy forces. Men, women, and children are desperately scrambling across to the other side of the bridge, their lives severely disrupted by this seemingly interminable conflict.
The old man's life has also been disrupted. He was forced to leave his hometown at short notice after it was evacuated by government troops. But war doesn't only have a negative impact on humans. The old man is worried about the fate of the animals he used to care for but which he was forced to abandon when he was ordered to leave San Carlos. He keeps telling himself that the animals will be alright, but that simply shows just how worried he is.
Aside from the destruction of life and human property, war can also destroy the human soul. That's what has happened to the old man; and that's why he sits down by the side of the road, unwilling to join the heaving mass of humanity hastily making its way across the bridge. His animals were his whole life; now that they've gone, he has nothing left. And so he awaits his fate. The destruction of his soul is a prelude to his physical destruction.