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?

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.