At the beginning of the passage in question, Rose hears her father, who is in his shed, say to himself "macaroni, pepperoni, Botticelli beans." Rose doesn't know what to make of her father listing these foods, but she has a vague impression that the "person who spoke these words and the person she spoke to as her father were not the same." From this quotation, we can infer that Rose is somewhat alarmed that the man she can hear in the shed, reciting a seemingly meaningless list of food items, seems to be so different from the the man whom she calls her father. In other words, Rose realizes that her father has a private identity all of his own, which he does not share with her.
Rose thinks that she can never ask her father about the words she hears him speak to himself in his shed, because to do so would be to make her father ashamed and embarrassed. She says that to ask her father about who he is inside his shed would be "the worst sort of taste" because she would be forcing her father to "acknowledge the person who was not supposed to be there."
Rose then hears her father, inside his shed, say something else. He says, "The cloud-capped towers, the gorgeous palaces." These words are from Shakespeare's The Tempest. Rose becomes alarmed either because she interprets the words to be nonsensical and thus an indication of her father's disconcerting eccentricity, or because she does not recognize as her father the man inside the shed who likes to read Shakespeare. Rose is thus alarmed by the idea that she does not really know who her father is. She is so alarmed that she describes hearing her father speak these words as "like a hand clapped against [her] chest, not to hurt, but astonish her, to take her breath away."
It is at this point that Rose feels that she has to run and "get away." She can't bear the thought of her father catching her listening to him. She thinks that this would be "terrible" perhaps because it would make him feel embarrassed, but most of all because it would be an irrevocable, mutual acknowledgement that Rose's father is not quite the man he has been pretending to be.