How do you recognize the story someone else is living? The answer is often to listen "as if" you were considering a play or movie. A play or a movie has a script: everything that is part of the script fits an overall narrative pattern. Particular phrases, characteristic responses and images are not just free-floating bits of information: in a script, each of these has meaning. Listening as if life were a script allows us to notice connections that we would otherwise miss.
The first step is to assign part of your mind to follow the conversation as if you were an observer and not an active participant. From this point of view, you can listen for recurring themes, words or images. As you listen, allow yourself to reflect what you notice - to repeat words, phrases or tones that seem to you significant. Do not worry that this influence direction as much as it follows it. There is no such thing as clean language: all language is part of a complicated interweaving of verbal and nonverbal influence. We always communicate and the people to whom we are talking always rely on our feedback to support their part of the connection.
The next step is to calibrate response to what you offer back, noticing when you 'hit a nerve,' and when the conversation follows the track you have suggested, and when it veers in a new direction. You will know what this means as it happens, and with practice, it will become natural for you to notice direction and pattern as easily as you now notice particular pieces of information.
You cannot adequately define what a story is (no one has), and yet you can tell stories and listen to them. You will not be able to adequately analyze another human being (no one has), and yet you can listen as if you knew what the story you hear means. The final step is to become part of the story: to respond not to the person, but to the story you have heard with story elements of your own. If the story ends with a wall, you can find a door or a window. If the story ends with tears, you can offer the thin edge of hope. If the story runs roughshod over others, you can be the voice of those the story has silenced.
You can do these things, not because you understand (you may not understand). You can do them because you intend to be both witness to and participant in the stories that other people are living.