The writer’s left a set of books? That’s significant (9)
I believe the answer is:
important
'significant' is the definition.
(synonyms)
'the writer's left a set of books?' is the wordplay.
'the writer's' becomes 'I'm' ('I am' referring to the writer of the clue).
'left' becomes 'port' (left side of a ship - opposite of starboard).
'set of books?' becomes 'NT' (New Testament).
'im'+'port'+'a'+'nt'='IMPORTANT'
'that's' acts as a link.
(Other definitions for important that I've seen before include "of great concern" , "Leading" , "Crucial" , "influential" , "It's vital" .)