Quieten beast of burden, very loud inside (6)
I believe the answer is:
muffle
'quieten' is the definition.
(muffling is a kind of quietening)
'beast of burden very loud inside' is the wordplay.
'beast of burden' becomes 'mule' (I've seen this before).
'very loud' becomes 'ff' (musical abbreviation).
'inside' is an insertion indicator.
'mule' placed around 'ff' is 'MUFFLE'.
(Other definitions for muffle that I've seen before include "Keep warm" , "Deaden(er of) sound" , "Stifle sound, deaden" , "Deaden, a sound or noise" , "Cover (to block sound)" .)