In "Pride and Prejudice", what is significant about Darcy's first visit to Lizzy at the cottage? Why does she refuse Mr. Darcy's offer of marriage?
What is significant about Mr. Darcy's first visit to Lizzy at the cottage is the fact that he is allowing his basic needs, his passion for Miss Bennett, to control his actions, rather than his usual logic and good judgment. He is so consumed with his desire for Lizzy that he blurts out his intentions like a boiling kettle giving off steam. He is physically relieved to have said what he said, clearly ashamed of his actions, displaying both contempt and desire for Lizzy. Instead of jumping at the chance to marry a really rich man, Lizzy rejects him and then insults him for speaking to her like she is some kind of fever that he cannot get over. It is as if Mr. Darcy is seeking a cure for an illness not a wife. He is rejected and resumes his gentlemanly stance, possibly being restored to his good senses. It is both comical and symbolic of both pride and prejudice, only now it is Lizzy who displays these qualities . She has switched places with Mr. Darcy so to speak, even though Darcy's prejudice is still present when he proposes to Lizzy.
I think you are referring to Darcy's visit to Lizzy at the Collins' cottage. What is significant here is that Darcy is seeking more information about Elizabeth, probing into her attitudes and her lifestyle. He makes a pointed remark about her "not always having been at Longbourn". As readers, we begin to understand that Darcy is separating Elizabeth from her family. He may disapprove of them, but he respects her.
Elizabeth refuses Darcy when he soon returns to propose because she is prejudiced against him. She has some realistic complaints about his behavior, and some complaints based on false information. In addition to this, Elizabeth is offended by Darcy's arrogant behavior. As she calls it, his "wilfull disdain for the feelings of others." She believes he is "ungentlemanly". It is altered behavior later, when they meet at Pemberly, that will begin to alter her opinion.
Mr. Darcy's first visit to Elizabeth signals a change in his attitude and behavior toward people of the lower classes specifically and toward all people generally. He tries to "lessen [her] ill opinion" of him by showing her that he is truly a man of generosity. Some critics suggest that this is Darcy's true nature, and his air of superiority is mistaken for his shy and reserved nature. Darcy recognizes that his upbringing has influenced him to behave badly in some cases, and he wants Elizabeth's forgiveness. He tells her, "By you I was properly humbled. I came to you without a doubt of my reception. You showed me how insufficient were all my pretensions to please a woman worthy of being pleased." His willingness to admit his flaws and his attempt to rectify them show his sincerity in trying to change.