Sort that's stylish provided inside (8)
I believe the answer is:
classify
'sort' is the definition.
('classify' can be a synonym of 'sort')
'stylish provided inside' is the wordplay.
'stylish' becomes 'classy' ('classy' can be similar in meaning to 'stylish'**).
'provided' becomes 'if'.
'inside' means one lot of letters goes inside another.
'classy' placed around 'if' is 'CLASSIFY'.
'that's' is the link.
(Other definitions for classify that I've seen before include "Arrange by category" , "Make secret for security reasons" , "Arrange in categories" , "Sort by category" , "Order" .)