Boaster's baronet trapping badger with fish (8)
I believe the answer is:
braggart
'boaster's' is the definition.
(synonyms)
'baronet trapping badger with fish' is the wordplay.
'baronet' becomes 'bt'.
'trapping' is an insertion indicator.
'badger' becomes 'rag' (both can mean to criticise).
'with' is a charade indicator (letters next to each other).
'fish' becomes 'gar' (type of fish).
'rag'+'gar'='raggar'
'bt' enclosing 'raggar' is 'BRAGGART'.
(Other definitions for braggart that I've seen before include "Show-off" , "One who boasts loudly" , "Big boaster" , "self-publicist" , "bigmouth" .)