。
She refused to talk about it any further.
Literal
She [topic-は] it [about-について] more-than-that talk [nominalizer-の] [object-を] declined.
断る is the everyday verb for declining or turning something down — invitations, requests, or, as here, the prospect of continuing a conversation. The verb 話す is nominalized with の so it can be marked by を as the direct object of the refusal: literally 'declined the talking-of-it.' それ以上 ('more than that') commonly pairs with refusal or negative expressions to mean 'any further,' and ~について sets the abstract topic of discussion. Phrased from outside, this reports the refusal as a fact rather than enacting it.