Why does Fitzgerald begin the story with what seems to be the end of a conversation then continue as Charlie walks into the bar several lines later?

Fitzgerald likely does this to inspire interest or curiosity in his readers. Sometimes, writers begin their story in the middle or even tail end of a conversation. They purposely provide no other information, which makes us curious about the significance of the conversation.
Our curiosity, of course, drives our desire to continue reading. Before long, we are engrossed in the story.
In Fitzgerald's "Babylon Revisited," he begins with our protagonist Charlie Wales asking after the welfare of different men. At this point, we don't realize that Charlie is talking to a barman at the Ritz Bar.
After the short, nonlinear conversation at the beginning of the story, Fitzgerald then provides a short background about the setting. Apparently, the story is set in Paris after the stock market crash of 1929. We learn that the once thriving French city is practically empty and that few wealthy Americans can be seen there. Fitzgerald then picks up the conversation with the barman, Alix. Earlier, we were privy to the tail end of Alix and Charlie's conversation.
Now, we learn that Alix has returned to Paris for a specific purpose. He wants to see his daughter, Honoria. From Alix's shocked "Oh-h! You have a little girl?" we begin to understand that Charlie's situation is more complicated than we first imagined. So, we read on, motivated by the desire to satisfy our curiosity. Certainly, the tactic of using piecemeal conversations to motivate curiosity is an effective one in the art of storytelling.