I think he begins the story with the dead animals as a way of signifying how death is such a big part of the story. Death is always in the background in this story, and sometimes it comes out to the foreground.
It is in the background as the men rehabilitate. They know they are lucky not to be facing it anymore. They form their groups based on who has really faced it (the narrator and the boy with no nose have not really, so they are less part of the group). Then it comes into the foreground when the major's wife dies. But it's been there all the time.
The major thought he could control his life, but death and other disasters are always there to destroy people's plans. I think that the dead animals symbolize this -- the people are no more in control than the dead animals were.
I believe you are referring to the epigraph in Hemingway's The Snows of Kilimanjaro. The epigraph situated directly under the title of Chapter One states:
Kilimanjaro is a snow covered mountain 19,710 feet high, and it is said to be the highest mountain in Africa. Its western summit is called the Masai “Nghe Nghe”, the House of God. Close to the western summit there is the dried and frozen carcass of a leopard. No one has explained what the leopard was seeking at that altitude. (Hemingway, Kilimanjaro, 1)
The leopard is generally regarded as the symbol of sloth, somewhat defined loosely nowadays as laziness. The animal also represents instinct, and primal behavior, without emotion or feeling. Since the animal is dead; its life is now over, especially at that altitude where everything including the air is pure, fresh and clean.
The protagonist, on the other hand, is lazy, emotional, erratic, and impulsive. He represents the opposite of what the leopard represents. Except for that fact that he was lazy and never did set out to do what he wanted to accomplish. His life ends tragically unfulfilled and dies thinking back on what his life would have been like if he had not been lazy and undeterred.