。
She gave him an excuse that worked against her.
Literal
She [topic-は] herself [against-に] disadvantageous-becomes [like-ような] excuse [object-を] him [to-に] gave.
~ような is the attributive form of ~ようだ; here it modifies a noun with the sense of 'an X like that, a sort of X that' — 自分に不利になるような口実 ('an excuse of the sort that works to her disadvantage'). 不利 ('disadvantage') + に + なる = 'become disadvantageous.' 口実 ('pretext, excuse') usually has a slightly negative tinge — an excuse offered to deflect blame. Pictures someone inadvertently handing the other person ammunition.