What it took me many years to realize is that the soloist is not the only one making dramatic things up on the spot. The band is actually re-harmonizing the tune on the fly, changing the original chords, often in extremely challenging and foreign ways. A passage might be shifted from a major to a minor key, in an extreme example. Soloing is the part of jazz that defines the identities of its greats. With a bit of careful listening, it is as easy to distinguish the saxophone of Charlie Parker from Coleman Hawkins as it is to distinguish between the voices of Frank Sinatra and Tony Bennett.
Great solos are like great speeches or sermons.
They make statements, embroider on them, reach out to a crowd and rouse them, quote from other sources, and develop ideas. They are personal and meaningful, and they only ask ears to hear. I recall having my breath taken away during a mid-1980s Wynton Marsalis concert by some uncanny ESP that passed between drummer Jeff Watts and pianist Marcus Roberts. Roberts' solo had built to a loud and busy crescendo, with Watts pounding away in full support. Suddenly, the thunder stopped, and both musicians dropped away to a peaceful little whisper of sound at exactly the same time, as if it had been composed by Stravinsky.