The notion that "Frollo is encouraged by the Archdeacon of Notre Dame to raise Quasimodo as his own, to atone for killing the baby's gypsy mother" on the stairs of the church had to be explained in the opening scene of the musical, and that was the catalyst for the song's creation.
[1] Disney Voice Actors: A Biographical Dictionary explains that "the opening sequence of [the film] was originally all narration and the result was deemed too lifeless so ['The Bells of Notre Dame'] was written".
During the song, Clopin tells a group of young children about the mysterious bell-ringer of Notre Dame Cathedral, described as "a tale of a man, and a monster."
Frollo pursues her on his horse, thinking her bundle contains stolen goods, in an intense chase through the streets that comes to a head on the steps of Notre Dame.
Frollo grabs the bundle from her, but in doing so strikes a blow to her head with his boot, causing her to fall down onto the stone steps, breaking her neck and killing her instantly.
Frollo callously tries to dismiss his actions, but the Archdeacon doesn't believe him and declares that no amount of lying will hide what he has done from "the eyes of Notre Dame" (belonging to the decorative statues built into the structure and used as a metaphor for God and His Angels).
Fearing for his immortal soul, Frollo nervously asks what he must do to atone for his sins and the Archdeacon replies that he must raise the orphaned child as his own.
The work Mouse Morality: The Rhetoric of Disney Animated Film by Annalee R. Ward argues that the use of a play-within-a-play technique used in this opening number "enables the filmmakers to condense some of the story, telling us the setting instead of showing it".
It adds that symbolically, Clopin's puppet show is a metaphor for what Disney has done to the original source material, having "reduced it to manipulated entertainment, ala 'tragedy lite'".