。
She's stuck on the idea that she's the best English speaker around.
Literal
She [topic-は] English [object-を] speak [nominalizer-の] [subject-が] number-one skilled [quotative-と] is-being-conceited.
自惚れる ('to be conceited / stuck on oneself') is built from 自 ('self') + 惚れる ('to fall for / be infatuated with'), so its image is literally 'falling for oneself.' The と here is quotative — it marks 英語を話すのが一番うまい as the content of her self-flattering thought. ~ている then puts that conceit into the present continuous: she's currently in the state of believing this about herself. 一番 + adjective is the standard superlative construction ('the most X / the X-est'). Note the chain of ideas: she speaks English (の-nominalized), that nominal is the thing-that-is-best (が-marked), and the whole judgment is what she vainly thinks (と).