Transported by an animal going around quietly (4)
I believe the answer is:
rapt
'transported' is the definition.
(I've seen this before)
'an animal going around quietly' is the wordplay.
'an animal' becomes 'rat' (rat is a kind of animal).
'going around' means one lot of letters goes inside another.
'quietly' becomes 'p' (p for piano and quiet).
'rat' going around 'p' is 'RAPT'.
'by' is the link.
(Other definitions for rapt that I've seen before include "Fully intent, fascinated" , "Ecstatic in part" , "Enthralled, captivated" , "Engrossed, spellbound" , "Spellbound, enthralled" .)