Anxiously is from anxious plus -ly. The suffix -ly recasts anxious as an adverb: the lexical idea remains recognizable, but the word now describes manner, degree, or circumstance rather than naming the quality by itself. In use as an adverb, anxiously denotes in a worried or nervous way. Typical contexts include “He anxiously awaited the arrival of his child”. The meaning stays close to the form: it applies the etymological material to in a worried or nervous way, keeping the visible parts legible in ordinary use.