How does the old man meets his impending death in "Old Man at the Bridge"?

In meeting his likely death, the old man is much more worried about the welfare of his animals than his own situation.
The story is told from the first-person point of view of a soldier whose task it is to investigate a bridge and then report the enemy's position. When he returns, he finds that one elderly man has been left behind.
Immediately after noting his town of residency, the old man mentions his animals. The narrator makes a vague, "Oh," comment and the old man follows up with "I stayed, you see, taking care of the animals." He lists the exact animals under his care and notes that he realizes that his cat can take care of itself but that he isn't sure what will become of the others. He asks the soldier a couple of times what he thinks will happen to the animals.
This is a man who is seventy-six years old and at this point in his life is "without politics." He has no family and nowhere else to go. The animals and his land were his sense of purpose; without them, he has nothing else.
And so he faces his impending death completely focused on the last meaningful connections he enjoyed in his life: his animals who depended on him.

Ernest Hemingway’s short story “Old Man at the Bridge” tells of the narrator’s encounter with an elderly man near a pontoon bridge over the Ebro River during the Spanish Civil War. The man appears to have no family or friends and is fleeing the conflict by himself.
The old man reveals that he was the last person to leave his village of San Carlos because he did not want to leave his animals. Caring for his cat, pigeons, and goats seemed to have provided the elderly man with a purpose for his life. Due to the fighting, the old man was forced to abandon his beloved animals. The Old Man fears for their safety amidst the shelling and fighting in his town. His morale is deflated by imagining his animals' demise.
When the narrator urges the old man to leave the pontoon bridge, he is too tired and uninterested in leaving. He remains seated in the dust, calmly resigned to his fate.
It is important to point out that Hemingway does not explicitly reveal the fate of this elderly man in his story “Old Man at the Bridge.” While it is likely that the man was killed in the ensuing fighting, it is not certain.
I hope this helps!