They have an exaggerated sense of their own importance (8)
I believe the answer is:
egotists
'they have an exaggerated sense of their own importance' is the definition.
(I've seen this before)
This is the entire clue.
(Other definitions for egotists that I've seen before include "Conceited and self-centered people" , "Self-centred self-seekers" , "People with exaggerated notions of their own importance" , "The selfish" , "They have an exaggerated sense of their own importance" .)