One of the lessons of conversation that I'm still working to get better at is how to let the person know I'm interested in what they are saying and invite them to add more instead of responding to the gap with a related story of my own.
I think part of it is my natural Midwestern tendency to think of asking direct questions as nosey. My instinct is to let them know I'm engaged in this topic which gives them space to share more if they want and stop where they are comfortable.