Demonstrates quietly and goes here and there (6)
I believe the answer is:
proves
'demonstrates' is the definition.
('prove' can be a synonym of 'demonstrate')
'quietly and goes here and there' is the wordplay.
'quietly' becomes 'p' (p for piano and quiet).
'and' says to put letters next to each other.
'goes here and there' becomes 'roves' (as in roving around).
'p'+'roves'='PROVES'
(Other definitions for proves that I've seen before include "Verifies" , "Provides evidence for" .)