How does Mary Shelley get us to sympathize with a monster? Where does she succeed and fail? Why does she do this?

Mary Shelley's Frankenstein creates moral tension between the two central characters: Victor Frankenstein, a young Genoese science student who has assembled and animated a creature using dead bodies, and the unnamed creature he creates. While the opening of the novel builds considerable animosity toward the character of the creature through the eyes of Victor, the creature's own description of his struggle to join humanity and his mistreatment at the hands of his creator and others leads to a more complicated portrait of his emotional landscape and moral choices. While the novel does not suggest that all his actions are defensible (for example, his murder of Victor’s younger brother is not excused), it does raise the question of the extent to which the trauma he has endured can make us sympathetic to the deep sorrows he expresses and the rage with which he responds.
Frankenstein is an epistolary novel (a novel told through letter writing) in which the stories of individual characters are often told through retellings by other characters in and beyond the letters of Captain Walton to his sister. When Walton first encounters Victor Frankenstein in the icy waters through which Walton hopes to discover passage to the North Pole, Frankenstein's own story comes to dominate the novel, and the frame narrative of Walton's expedition moves to the sidelines. As a result, the early descriptions of the "monster" Victor has created are told only by Victor himself, whose disgust with the creature becomes the reader's first emotional barometer for evaluating the character.
This changes dramatically in chapter 10, in which Victor encounters his creation. While Victor expresses rage at the murders the creature has committed, the creature, in turn, demands that Victor hold himself accountable for his choice to abandon his own creation upon first sight. In a directly stated allegory for human paternal abandonment, the creature tells the story of his own confused and terrified quasi-infancy, in which, without his creator, he fumbles into the nearby forest alone:

It was dark when I awoke; I felt cold also, and half frightened, as it were, instinctively, finding myself so desolate. . . . I was a poor, helpless, miserable wretch; I knew, and could distinguish, nothing; but feeling pain invade me on all sides, I sat down and wept.

Upon encountering this, the novel's reader—perhaps for the first time—is asked to evaluate the impact of Victor's abandonment on his vulnerable pseudo-child.
The novel further makes room for sympathy for the creature as he discovers the dangerous way humans respond to his physical body, whereupon he hides in the cottage of a French family: a brother and sister named Felix and Agatha; their aging father; and, later, Felix's wife. As he slowly learns language by eavesdropping on their conversations, the creature develops a strong attachment to the family. He begins to crave acceptance from them, and fantasizes that he can earn—through hard study—a place in their hearts:

My thoughts now became more active, and I longed to discover the motives and feelings of these lovely creatures; I was inquisitive to know why Felix appeared so miserable and Agatha so sad. I thought (foolish wretch!) that it might be in my power to restore happiness to these deserving people. . . . I formed in my imagination a thousand pictures of presenting myself to them, and their reception of me. I imagined that they would be disgusted, until, by my gentle demeanour and conciliating words, I should first win their favour and afterwards their love.

This combination of intellectual maturation and deep affection for the family introduces to the reader a level of emotional complexity in the creature which contradicts the earlier portrayals of him as wholly demonic. In chapter 15, when the creature is attacked by the family during his attempt to bond with them, his initial pain at being abandoned by Victor is redoubled. Using the notebook he found among Victor's possessions in his first hours of life, the creature begins to plan systematic and murderous revenge against his creator.
This turn toward violence, while not exonerated by the novel, merits evaluation in the context of the creature's fuller emotional landscape. The novel ends with the creature, in the wake of Frankenstein's death, expressing deep remorse for his actions and entangled in his complicated love for Victor and anger toward himself:

But it is true that I am a wretch. I have murdered the lovely and the helpless; I have strangled the innocent as they slept and grasped to death his throat who never injured me or any other living thing. I have devoted my creator, the select specimen of all that is worthy of love and admiration among men, to misery; I have pursued him even to that irremediable ruin. There he lies, white and cold in death. You hate me, but your abhorrence cannot equal that with which I regard myself. I look on the hands which executed the deed; I think on the heart in which the imagination of it was conceived and long for the moment when these hands will meet my eyes, when that imagination will haunt my thoughts no more.

The novel does not neatly tie up the moral conclusion of this complicated relationship. Both characters leave the story in the novel's final moments (one through death and one through self-imposed exile), unable to undo the choices they have made and anguished by the consequences of those choices. In this way, in both their sympathetic attributes and their unforgivable ones, Victor and the creature end the novel most closely resembling one another.