The shirts symbolize the way Gatsby is trying to impress—to buy—Daisy with his wealth. He believes that his money makes him worthy of her love. As he tosses out beautiful shirt after beautiful shirt in all the colors of the rainbow, he is signaling to her that he can now meet her in terms of affluence.
The whole encounter has been carefully staged from the start to impress Daisy. Gatsby wants to meet her in Nick's more modest home so that he can wow her with his mansion only after the first shock of their reunion has ended. He has Nick's lawn mowed before Daisy comes. He quickly takes her on a tour of his grand mansion. With the shirts, Gatsby, who was impressed with the careless ease with which the younger Daisy took her affluence for granted, can finally offer her what appears to be limitless wealth.
Of course, the efforts he goes to and the way he throws out all his shirts before her show that wealth will never come effortlessly to him. He will always have to show it off to confirm through other people's eyes that it is real. He will always try too hard.
Daisy is overwhelmed, as her crying into the shirts indicates, and also not in touch with her feelings, as her inability to explain her crying suggests. By asking her to accept him on the basis of his wealth, on what he can offer her materially, Gatsby does show he understands her. Ironically, however, he is competing on a field where he can't win: Tom's old money and higher social status will always be more attractive to Daisy.