。
She jumped at the chance to go to New York.
Literal
She [topic-は] New-York [to-に] go chance [at/onto-に] jumped.
飛びつく is a compound verb 飛ぶ ('fly, jump') + つく ('attach, cling'), giving 'jump at, leap onto, eagerly seize.' Combined with チャンス ('chance, opportunity') it forms a near-perfect calque of the English idiom 'jump at the chance.' The に here marks the target of the jumping — what one is jumping at, not where one is jumping. ニューヨークに行くチャンス is a relative clause modifying チャンス: 'a chance [that involves] going to New York.' Note how Japanese stacks the modifier before the noun, the opposite of English's 'a chance to go to New York.'