The charge develops when the lattice structure is stressed. At the atomic level, valence band electrons become conduction band (free) electrons at that point (yes, a quantum leap occurs), hence a charge (negative) builds up, and the surface of the piezo becomes more conductive. When the stress is removed, the piezo reassumes its intrinsic insulator properties.
From the piezo, the potential difference can be detected in more than one way. What I described is but one (common) way the changes can be sensed.
The phenomenon you are describing is called the 'direct piezoelectric effect'. Vibrations from the strings cause the stress, and therefore the changes in potential.
The 'converse piezoelectric effect' is exactly the opposite mechanism. Applying an electric field to the piezo causes physical vibrations in the ceramic. We see this phenomenon in the piezo tweeter.