Starts to enjoy drama gripping Edinburgh Fringe (4)
I believe the answer is:
edge
'fringe' is the definition.
(synonyms)
'starts to enjoy drama gripping edinburgh' is the wordplay.
'starts to' indicates taking the first letters.
The initial letters of 'enjoy drama gripping edinburgh' is 'EDGE'.
(Other definitions for edge that I've seen before include "Outer limit" , "Border, fringe" , "Create borders" , "'Margin, rim (4)'" , "Rim,border" .)