Read the passage that begins "Yes, Miss Louisa—they always remind me of stutterings" and ends "'Merrylegs;' she whispered the awful fact; 'is his performing dog'" (book 1, chapter 9). How does Dickens make the extract a moving one?

Dickens makes this passage a moving one by focusing on Sissy Jupe as an emotional being. In the Gradgrind world of utilitarianism and rationality that she has entered, Sissy thinks primarily with her heart and her imagination, and though the school has already taught her that this is very, very wrong, she fails at aligning herself with its way of thinking. By the time we are done with this passage, we, as readers with hearts, are on her side.
In this passage, she is speaking with Louisa about the problems in school that make her "low spirited." She has already revealed her empathy with the starving, and now she tells Lousia that she gave the wrong answer when she was asked at school to provide the percentage if "only" five hundred people of a hundred thousand died at sea. She replies with the answer, "nothing." She is thinking with her heart, not her head, because by "nothing" she means that cold percentages mean nothing at all to the heart-broken relatives of the people who were killed. In making statements like that, Sissy is unwittingly showing the hard-heartedness of Gradgrind's "greatest good for the greatest number" utilitarian philosophy. She is pointing out the weakness of utilitarianism, which is that it cuts the human sympathy and imagination out of life. Sissy, part of the magical and imaginative world of the circus, places putting herself into the skin and heart of people who are suffering ahead of treating them as statistics.
Louisa goes on in this passage to coax out of Sissy the story of her father, who has deserted her. This kind of personal, emotion-based conversation that veers away from facts and figures is forbidden, but Sissy nevertheless moves us by speaking of her love and compassion for her father. She tells Louisa he was a clown, though she has been taught by Gradgrind that being a clown is a useless thing, and that though he made people laugh he was also a sad, despairing man. She raises our sympathy too in revealing he was treated cruelly:

But they wouldn’t laugh sometimes, and then father cried. Lately, they very often wouldn’t laugh, and he used to come home despairing. Father’s not like most. Those who didn’t know him as well as I do, and didn’t love him as dearly as I do, might believe he was not quite right. Sometimes they played tricks upon him; but they never knew how he felt them, and shrunk up, when he was alone with me. He was far, far timider than they thought!

We also are moved to find out that Sissy lost her mother as a baby, and that Sissy would comfort her father at night by reading him the kind of romantic fiction that Gradgrind abhors as useless, such as the Arabian Nights:

he used to forget all his troubles in wondering whether the Sultan would let the lady go on with the story, or would have her head cut off before it was finished.

We are moved by Sissy's innocent love for and loyalty to her father.
In sum, Dickens uses sentiment, or the reliance on feelings, to move us as readers. He does this by showing Sissy putting compassion for individuals ahead of facts and figures and by revealing her courage and caring in struggling bravely through a hard existence without a mother and with a depressed, despairing father. Dickens knows that details convey emotions, and so he gives us the details of Sissy's life. Rather than telling us that utilitarianism is a harsh, dehumanizing philosophy, he shows us through Sissy's caring, kind heart what is missing when life is reduced to statistics, which Sissy rightly calls "stutterings." In fact, they mean nothing against lived human experience, relationships, imagination, and love.