One of the defining characteristics about Joyce's work in Dubliners is that each story features the main protagonist having an epiphany, or sudden insight or realisation about himself or herself. You might want to compare and contrast the epiphany in "The Dead" with another story such as "Araby" for example to explore this element of Joyce's fiction.
However, consider how this epiphany functions in "The Dead." We are presented with a central character who seems overly preoccupied with how others view him. Note how he responds to the rebuff made by Lily as she takes his coat:
Gabriel coloured as if he felt he had made a mistake and, without looking at her, kicked off his goloshes and flicked actively with his muffler at his patent-leather shoes.
Even though no blame is apportioned directly, he feels self-conscious enough to consider he has done something wrong. Then, rather than ask for her apology, he then insultingly gives her a gift of money, excusing it as a present for Christmas. We see this aspect of Gabriel again and again throughout the tale, a concern about how he looks and what others think of him and his inability to know how to patch up mistakes.
However, it is only at the end of the story, when he has found out about the existence of his "rival," Michael Furey, that he is able to see himself and his relationships with others, and he experiences an epiphany about who he is:
Generous tears filled Gabriel's eyes. He had never felt like that himself towards any woman but he knew that such a feeling must be love. The tears gathered more thickly in his eyes and in the partial darkness he imagined he saw the form of a young man standing under s dripping tree. Other forms were near. His soul had approached that region where dwell the vast hosts of the dead.
Gabriel realises that he has never truly loved, in comparison with Michael Furey, and he realises in addition that it would have been "Better to pass boldly into that other world, in the full glory of some passion, than fade and wither dismally with age." His inability to experience and feel such emotions deeply means that he falls into the latter category.