Beast that's fussy I'm breaking in (6)
I believe the answer is:
animal
'beast' is the definition.
(a beast is an animal)
'fussy i'm breaking in' is the wordplay.
'fussy' becomes 'anal' (anal can mean overly fussy or orderly).
'breaking in' is an insertion indicator.
'anal' enclosing 'im' is 'ANIMAL'.
'that's' is the link.
(Other definitions for animal that I've seen before include "swine" , "Mammal" , "Source of meat" , "Circus performer, maybe" , "Man or beast" .)