。
She admitted she can't speak French.
Literal
She [topic-は] French [object-を] cannot-speak [thing-こと] [object-を] admitted.
喋れない is the negative potential of 喋る ('to speak, chatter') — a slightly more casual or active verb than 話す. The admitted content ('that she can't speak French') is wrapped with こと into a noun phrase serving as the object of 認める ('admit, acknowledge'). The grammar of admitting in Japanese is reliably this こと-を-認める shape: the fact admitted gets nominalized with こと, then takes を as the direct object of the admission. The construction is identical whether one is admitting a small fault or a serious crime — only the content changes.