。
She can speak French fluently.
Literal
She [topic-は] French [object-を] fluently speak [thing-事] [subject-が] can-do.
流暢 ('fluent, eloquent') is a Sino-Japanese compound — 流 ('flow') + 暢 ('extend, smooth'), unified by the metaphor of speech flowing without obstruction. 流暢に is the adverbial form. The 事 in 話す事ができる is the same nominalizer こと, written here in kanji rather than the more common hiragana — a stylistic touch that gives the sentence a slightly stiffer, more written feel.