Family time in spot that's fittingly close (4-5)

'fittingly close' is the definition.
(skin-tight describes tight-fitting clothing)

'family time in spot' is the wordplay.
'family' becomes 'kin' (synonyms).
'time' becomes 't' (abbreviation).
'in' means one lot of letters goes inside another.
'spot' becomes 'sight' (both can mean to notice or observe).
'kin'+'t'='kint'
'kint' inserted inside 'sight' is 'SKIN-TIGHT'.

'that's' is the link.