She declared firmly that she would not go to the movies with him.
Literal
She [topic-は] he [with-と][contrast-は] movie [to-に] not-go [quotative-と] firmly said.
彼とは uses contrastive は added to the companion-marker と — 'with him (specifically, as opposed to others).' This emphasizes that the refusal targets him in particular. きっぱり is a 擬態語 (state mimetic) for decisive, firm action. The whole quoted clause is reported via と + 言う.