Miss Wilkinson's "Swan Lake" story is significant for a couple of reasons.  The first is that it establishes an artistic frontier for Billy.  In exploring the Swan Lake story, she is able to establish in his mind an artistic metric significant enough to motivate him to be the best dancer he can be.  The Swan Lake story and its summative meaning to Billy is more than simply a story.  It represents an artistic benchmark, something seen accomplished at the end of the film.  

At the same time, the Swan Lake story is meaningful because it represents how Miss Wilkinson sees Billy.  The swan's transformation into a human being for a few hours is representative of Billy's feeling when he is dancing. She is able to keenly recognize that in Billy lies the symbol of the swan.  The transformation that happens for just a few hours for the swan enables transcendence in a world of contingency. In much the same way, Billy experiences this same transcendence when he dances.  As he describes it as "electricity," there is a small window of time when Billy is able to move beyond the context and contingency in which he lives.  This is another instance in which the story has direct and meaningful connection to Billy.  Accordingly, Miss Wilkinson relays the story for just as the swan will die if it is not rescued by the prince, Billy's soul will perish if dance is not a part of his life.  

To rescue Billy, the embodiment of the swan, specific people play vital roles. Their roles enable him to stay true to his struggle for his gift.  One such individual would have to be Miss Wilkinson.  She is the first to recognize the talent that lies within him.  Miss Wilkinson clearly understands how dance can be liberating for Billy.  She encourages him on artistic levels as well as on a spiritual level, as she is able to help to convince him that dance is his calling.  In a world where Billy is besieged by forces that seek to repudiate his bond with dance, she seeks to cultivate it.  At the same time, I would suggest the Billy's father ends up playing a vital role in furthering his son's gift.  He is committed to his son's talent when he sees Billy dancing.  The commitment he shows towards Billy helps to inspire his son.  Being willing to cross the picket line and finance Billy's audition in London by pawning his mother's jewels are examples of how he ends up becoming a force that helps Billy achieve his dream, and help Billy stay true to his goal.  In seeing his father's example, Billy becomes galvanized to pursue his dream and find success in it.  At the same time, the neighborhood miners and people who help to raise money for Billy also demonstrate the need for Billy to stay true to his goal.  Billy's quest starts off as a solitary one. Yet, by the end of the film, it is one in which others have joined for they have recognized the authenticity of his talent and the need to validate it.