。
My mother absolutely hates writing letters.
Literal
Mother [topic-は] letters [object-を] write [nominalizer-の] [subject-が] greatly-dislike [explanatory-なんです]
手紙を書くのが大嫌い = 'greatly dislikes the act of writing letters.' の nominalizes the verb clause; が marks it as the subject of 大嫌い. なんです = colloquial のです — adds an explanatory/emphatic nuance.