The officer looked at Daisy while she was speaking, in a way that every young girl wants to be looked at sometime, and because it seemed romantic to me I have remembered the incident ever since.
This is a clever work of sentence construction – if Fitzgerald had tried to describe what Gatsby looked like as he gazed at Daisy, the description might have worked for some readers but fallen flat for others. So instead the author describes the look as a fulfillment of universal fantasy. This breaks the hackneyed writing rule of “show, don’t tell” – but that rule was always more of a Hemingway thing than Fitzgerald.
Romance is better described by its effect than its actions, because romance is ultimately so personal that the actions that seem romantic to one person can seem ridiculous to everyone else. What is the way that every young girl wants to be looked at? Try to describe any one way, and a thousand young girls will say, “Eeeew.” But it’s close enough to the truth to say that every young girl imagines that there is a special way that no one has ever looked at her before. She herself couldn’t put it into words, other than the words for how that look should make her feel. Jordan remembers the incident not for the way that Gatsby looked at Daisy, but for the effect upon Daisy.