A convincing interpretation of an improvisatory passage may at first allude us because we may be looking for something more complicated than what it is actually there. Just because there are gobs of notes does not necessarily mean there are more than a few underlying ideas. How I teach piano students the written-in improvisatory gestures in repertoire depends on whether they have already improvised or composed. This requires that both the student and teacher have a willing imagination, a desire to continually improve the ear, good analytical skills, and a bit of an adventurous spirit. I first use guidance teaching to evoke comprehension of the objective aspects of the passage.
Aural recognition is best, but visual and intellectual skills are also usually needed.
As they do so, they experience the inherent precariousness that looms during the birth of an idea. Then I have them apply this same feeling to Mozart's initial motif in this cadenza. This might result in their playing it in a sneaky or tentative way, or even in a spirit that I didn't anticipate - that's good, too. If it makes any sense at all, I encourage it. Its fresh sound is the impetus for the melody suddenly changing direction.