What is the signifigance of the ending to The Phantom of the Opera?
Your question is a little vague. There are in effect two endings to the "story," and you don't indicate which "ending" you are asking about. The reason there are two endings is that the story of "the Opera ghost" is a frame story.
The narrative begins with the frame in which the first narrator (a first-person narrator), singing himself as Gaston Leroux, tells about documents he has received and confirmed the truth of that tell an eye-wittenss account of a whole chain of events that were witnessed by or told to the Persian.
After the Prologue--in which the first narrator, Leroux, speaks--the story unfolds from "Chapter I Is it the Ghost?" as the second first-person narration, the Persian himself, described events in the documents he handed over to Leroux:
The Persian, second narrator: The shyness of the sailor-lad—I was almost saying his innocence—was remarkable. (Chapter II The New Margarita)
The premise set up in the Prologue is that Leroux is recasting the documented events as the Persian saw them or learned about them from first-hand participants, like Christine.
... when the Persian had told me, with child-like candor, all that he knew about the ghost and had handed me the proofs of the ghost's existence—including the strange correspondence of Christine Daae—to do as I pleased with, I was no longer able to doubt.
Thus the final ending is the completion of Leroux's frame narration: he provides the epilogue that leads us to the skeleton of the Phantom. The significance of this ending is that we learn the whole backstory of Erik's life and crushed feelings; we learn about his final end because, though a "monster," we feel sympathetic toward "the Opera ghost"; and we are impressed by the "truth" of the whole narrative in keeping with the literary convention of wrapping fiction in a nonfiction frame.
The ending of the Persian's story (the first ending) sets Christine and Raol free and delivers Erik over to the imminent death of a broken heart:
Erik bowed his head and said:
"I have not come here ... to talk about Count Philippe ... but to tell you that ... I am going ... to die..."
"Where are Raoul de Chagny and Christine Daae?"
"Of love ... daroga ... I am dying ... of love ... That is how it is ... loved her so! ... And I love her still ... I tell you!" [...]
"I went and released the young man," Erik continued, "and told him to come with me to Christine ... They kissed before me in the Louis-Philippe room ... Christine had my ring ...."
The significance of this ending is that we learn of the fates of the beloved Christine and Raol ... and we learn whether Erik does have a human heart that still beats with compassion and true love in a breast overpowered by a monster's face. We learn that, yes, he does and we mourn for his death.
Have you ever heard the expression, "If you love it, let it go"? This is certainly the significance of the ending of the story of the Phantom. Erik, the Phantom, is madly in love with Christine. He gives her a final option--to choose between him and Raoul. Christine does know that if she chooses to reject Erik, most of Paris will be blown up.
"If you turn the grasshopper, mademoiselle, we shall all be blown up. There is enough gun-powder under our feet to blow up a whole quarter of Paris. If you turn the scorpion, mademoiselle, all that powder will be soaked and drowned."
She understands Erik's temperament and decides to save Paris. Erik floods the basement of the Opera House, where Raoul and the Persian are trying to escape, to drown the gunpowder. At the last minute he has a change of heart because Christine begs him and devotes herself to him, as "my living wife," and he saves both men who are trapped. After both recover, he lets Christine see Raol and watches their kiss. This is when he lets Christine go with Raol. Erik realizes that Christine cannot love him while she loves Raoul. So, when he allows Christine and Raoul to run away together, he asks her to come back and bury him after he dies, telling where to find his body and where to secretly bury him after putting the gold ring on his finger.
"I made Christine swear to come back, one night, when I was dead, ... and bury me in the greatest secrecy with the gold ring, ... I told her where she would find my body and what to do with it... if Christine keeps her promise, she will come back soon! ..."
Later, as Erik requested, the Persian posts an obituary in the paper signifying Erik's death. As he claimed to the Persian he really was dying and the narrative of the Persian concludes with Erik's death.
Lastly, Erik relied on the Persian, ... to inform the young couple of his death and to advertise [his death] in the EPOQUE.
The Persian had seen the poor, unfortunate Erik for the last time. Three weeks later, the Epoque published this advertisement:
"Erik is dead."