Many great musicians and music teachers have described good music as storytelling. A story is a series of events that usually involve overcoming an obstacle. Great stories set up a scenario and then surprise us with the ending-boring stories don't. People from every culture love stories, whether sharing what they learned or following the exciting journey of a certain character.
The musical equivalent of a character is a motif, which is a short musical idea or a theme. Instead of a character going through various scenes in a movie, a motif can move through various chord progressions in a tune. The character moves through the story -- whether in music or in a movie -- by what decisions they make. Improvisation is the manifestation of those decisions.
Improvisation as a concept is something all humans do all the time. Every time you have an option for behavior in the moment and you take one instead the other -- that’s improvisation. You are improvising when making a decision. There are thousands of these little decisions in a jazz solo: what pitches to play, and what rhythms, rests, contours, attacks and dynamics to use.
When the character(s) in your solo make surprising decisions and actually go somewhere -- now your story is progressing.
Many people think that just because they are playing they are telling their story. That may be true if your story means you are playing what's familiar to you -- that is the equivalent of taking the audience to your front porch. If you can tell a musical story that takes the musician and listener to unfamiliar territory, a surprising panorama and to several destinations (and back again), your musical story will be compelling to musicians and civilians alike.