Braggarts removing small hats (7)
I believe the answer is:
boaters
'hats' is the definition.
(boater is a kind of hat)
'braggarts removing small' is the wordplay.
'braggarts' becomes 'boasters' ('boaster' can be a synonym of 'braggart').
'removing' indicates named letters should be taken away.
'small' becomes 's' (abbreviation - e.g. clothes size).
'boasters' with 's' taken away is 'BOATERS'.
(Other definitions for boaters that I've seen before include "top gear" , "Thames travellers" , "straw hats?" .)