The main character, or protagonist, of a story is the character who undergoes change as a result of the story's conflict. A conflict is a struggle between two opposing sides in a story, usually described as a struggle between the protagonist and the antagonist. However, an antagonist does not necessarily have to be a character. The protagonist can also struggle against society, nature, and even the protagonist's own self. So, to identify the protagonist, we identify the character who changes as a result of the conflict.
Interestingly, in Sherwood Anderson's short story "The Egg," the nameless narrator's father experiences conflicts between self and fate. He had at first been very happy as a farmhand; however, his wife persuaded him to try and be more financially successful by starting a business of his own. Instead of being true to himself, the narrator's father bowed to his wife's wishes, showing us the father's conflict with his own self. As a result, the father endured failure after failure, and since he had no control over his failures, we can see these failures as fated. Hence, the second conflict the father endures is against fate.
Yet, though the father encounters these conflicts, he actually does not change or grow as a character as a result. Instead, he seems to become resolved to just give up. It is actually the narrator who undergoes changes. By the end of the story, due to witnessing his father's struggles, the narrator has developed a new perspective on life that stays with him for the rest of his life: He develops the very existential perspective that life is completely hopeless and meaningless, nothing but an endless cycle of futility, as we see in one of his final sentences, "I wondered why eggs had to be and why from the egg came the hen who again laid the egg."
Hence, even though the father struggles against the most apparent conflicts, it is actually the narrator who changes as a result of the conflicts, making the narrator the protagonist rather than the father. We can also see the narrator experiencing his own conflicts against self, life, and fate if we observe him being severely affected by his father's own struggles that he witnessed.
A final clue that the narrator is the protagonist concerns the fact that the story is written in the first person point of view, allowing is us to view the story through the eyes of the narrator, not the father. The story also begins and ends with the narrator, not the father.