Giggle about wife on social media site (7)
I believe the answer is:
'social media site' is the definition.
(Twitter is a social media website)
'giggle about wife' is the wordplay.
'giggle' becomes 'titter' (synonyms).
'about' is an insertion indicator.
'wife' becomes 'w' (abbreviation).
'titter' enclosing 'w' is 'TWITTER'.
'on' acts as a link.
(Other definitions for twitter that I've seen before include "Utter tremulous sounds" , "Sound made by bird" , "e-fad" , "Palpitate" , "Social networking site" .)