"How does the creature persuade us or Victor to hear his story?" Regardless, the answer may be the same.
The creature, like most of the characters in the novel, uses dramatic monologue, or long speeches filled with emotional and poetic language. This, of course, is not how people talk, but Shelley's elevated style breathes life into the monster. Despite his hideous appearance, the monster learns to become human first and foremost through language.
The creature's appeals to Victor center around identity and existence: he wants to know who he is and why he was created. Since he is the only one of his kind, he wants his creator to make him a mate; like Adam in the Garden, he is lonely. His dramatic monologues are thus filled with the rhetorical appeal of pathos (emotional arugment) aimed at Victor and the reader to empathize with his existential confusion:
How can I move thee? Will no entreaties cause thee to turn a favourable eye upon thy creature, who implores thy goodness and compassion? Believe me, Frankenstein: I was benevolent; my soul glowed with love and humanity: but am I not alone, miserably alone? You, my creator, abhor me; what hop can I gather from your fellow-creatures, who owe me nothing? They spurn and hate me. The desert mountains and dreary glaciers are my refuge. I have wandered here many days’ the caves of ice, which I only do not fear, are a dwelling to me, and the only one which man does not grudge. These bleak skies I hail, for they are kinder to me than your fellow-beings. If the multitude of mankind knew of my existence, they would do as you do, and arm themselves for my destruction.
Shelley's husband was a poet, and she knew how to fill her prose with poetic conventions. Notice the imagery, the apostrophe (address to a person), rhetorical questions, and alliteration. Such poetic rhetoric is enough to illicit sympathy from even the cruelest hearts.