Nobody knows for sure.
The Greek phrase is “ἔπεα πτερόεντα”, and “winged words” is a literal translation. The idiomatic meaning of this expression is not known, and it has spurred considerable debate amongst translators and scholars.
Herbert Jordan, who translated Homer into English, shares some of the issues he encountered on his website. He discusses winged words.
A common school of thought is that “winged words” connote speed in some manner — either emphasizing the spontaneity of the words, or indicating that the words were spoken quickly. This interpretation is found amongst ancient and modern studies¹. George Calhoun contended that winged words were spoken with unusual emotion or intensity. At the other end of the spectrum his student Milman Parry held that the words held no particular meaning and that Homer “uses this phrase just because it is useful, and without thought for any particular meaning which the epithet ‘winged’ might have”. Winged words played an important role in the elaboration of some theories about oral traditions.
Some translators have translated the phrase literally, others have reflected a perceived emotion, yet others ignored these words.
Incidentally, the expression “winged words” has come to mean a phrase that started as a quote but then took a life of its own. The very coining of this usage by Georg Büchmann made “winged words” winged words.
¹ F. M. Combellack, Words that Die. The Classical Journal, 1950.
² M. Parry, About Winged Words. The Classical Journal, 1937.