Daisy is representative of the upper class. One of her upper-class traits is that she does not show her feelings. We sense, however, that she has very strong feelings and is extremely sensitive. During her visit to Gatsby's mansion, Daisy gradually realizes how important she is to him. All of this material splendor has been gathered for her sake. Because she is truly upper class, she clearly sees the "new money" tastes represented by everything Gatsby has acquired.
When she buries her face in Gatsby's beautiful shirts and starts to sob, it is like a culmination of her feelings about everything Gatsby has shown her up to that point, including his pretentious mansion itself with its Marie Antoinette music rooms and Restoration salons. She is sobbing because she knows he has only been thinking of pleasing her but lacks the style, tastes, manners, and everything else people of her social class take for granted. The shirts are excellent symbols of Gatsby's naïveté and gaucherie. They are not only gaudy, expensive, and superfluous, but it is in ridiculously bad taste for Gatsby to display them to her like a grand finale.
He took out a pile of shirts and began throwing them, one by one, before us, shirts of sheer linen and thick silk and fine flannel, which lost their folds as they fell and covered the table in many-colored disarray.
Daisy's feelings are too complicated to describe. She loves Gatsby but can see his shortcomings, including the fact that he couldn't possibly understand how she feels about his display of his shirt collection. Her feelings overwhelm her, and she bursts into tears, attributing them to being overwhelmed by the spectacular display of shirts.