A very excellent question! What this whole scene really comes down to is the theme of value. Edward valued loyalty above material gain and even above the good opinion of his relations. He even viewed loyalty as more important than his own personal preference or happiness.

But to apply it to the themes of attachment versus connection, you have to understand the characters' perspectives. Edward was choosing the right course—that of marrying Lucy even though it meant losing his relatives' esteem, his own inheritance, and even the chance of ever being with the person he really loved, Elinor. He viewed his attachment to Lucy as a binding one; he had promised himself to her and was going to live by that promise, despite everything it would cost him. His family viewed connection as a greater importance—thus the splitting of Edward from his family.

Lucy is another matter altogether; engaged to Edward at a young age, it is debatable whether or not she ever actually loved him. Perhaps at first, she felt the same infatuation that Edward felt for her. However, by the time we meet her, it is evident that her attachment is more to Edward's position and wealth than him as a person. She, like Mrs. Ferrars, values connection above attachment, and this is proved when she later leaves the disowned and disinherited Edward for his wealthy, fashionable brother.

Elinor, on the other hand, respects Edward all the more for his proving true to his promise to Lucy, even though it would forever take him from her. Her attachment to Edward is not in words; it is of the heart. Nevertheless, she values, as does Edward, the quality of loyalty and faithfulness, even at the cost of one's own happiness.

I hope this answers your question!