"to soar through air; move through the air with wings," Old English fleogan "to fly, take flight, rise into the air" (class II strong verb; past tense fleag, past participle flogen), from Proto-Germanic *fleugan "to fly" (source also of Old Saxon fliogan, Old Frisian fliaga, Middle Dutch vlieghen, Dutch vliegen, Old High German fliogan, German fliegen, Old Norse flügja), from PIE *pleuk-, extended form of root *pleu- "to flow."
Meaning "go at full speed" is from c. 1300. In reference to flags, 1650s. Transitive sense "cause to move or float in air" (as a flag, kite, etc.) is from 1739; sense of "convey through the air" ("Fly Me to the Moon") is from 1864. Related: Flew; flied (baseball); flown; flying. Slang phrase fly off the handle "lose one's cool" dates from 1825.