That man's Conservative background (7)
I believe the answer is:
history
'background' is the definition.
(I've seen this before)
'that man's conservative' is the wordplay.
'that man's' becomes 'his' (belonging to a man).
'conservative' becomes 'tory' (Tory means a member of the Conservative Party).
'his'+'tory'='HISTORY'
(Other definitions for history that I've seen before include "Time passed" , "eventful past" , "Chronicle" , "Background" , "'... repeats itself'" .)