。
She refused to go with me.
Literal
She [topic-は] me [with-と] go [nominalizer-の] [object-を] refused.
断る ('to refuse, to decline') takes a nominalized verb phrase as its object via のを. The の here turns the verb phrase 私と行く ('go with me') into a noun-equivalent that can take the object marker を. This Xのを + verb construction is the standard way to apply a verb to an action: Vのを忘れる ('forget to V'), Vのを楽しむ ('enjoy Ving'), Vのをやめる ('quit Ving'). 断る has connotations of polite refusal — declining an invitation or offer rather than blunt rejection.