Arching over many extant Egyptian
papyruses,
carvings, and other
media, this symbol must have been very important to the scribes and priestly caste of ancient
Egypt. As presented in color it is a disk the same shade of red as was used for portraying male skin, with two uncolored, purple, or blue many-feathered
wings spread wide to each of its sides. Where the disk meets the wing, a
uraeus, the symbolically important
cobra in striking position, is often affixed on each side. It is always presented grandly on the medium in question, overscoring either the entire piece or the central
text area or
illustration.
Sun worship could be considered the basis of most eras of the Egyptian state religion, with a different god and mythos for all of the sun's positions: Akhet / Harmakhet, god of the horizon and rising sun; Khepri, the scarab representing morning; Khnum, the ram-headed god of evening; and of course Ra and Amun-Ra, and later Horus, the sun's disk in its full glory in the midday sky. Thus, the winged sun is a representation of Horus, and its wings are the wings of the falcon, Horus's animal representation, keeping his disk aloft in its journey across the sky. This is the easy answer, and the explanation that has been used in Egyptology by default.
Of course, there is more to it than this. The Egyptians were highly conscious of what was above them, their flat desert plains and dry air making an environment perfect for watching the sky. At different points in their 2500+ years of history they understood the movements of four of the planets, used Sirius's position in the sky to keep track of passing years, and could use Polaris to orient their pyramids perfectly with the four compass points. All of this has made it extremely confusing that there isn't a single mention of a solar eclipse in their almanacs, a single depiction of such a rare and awesome event occurring in the sky. Unless there is one, an illustration not recognized until fairly recently even after hundreds of years of research into Egyptian history: the winged sun symbol itself.
The sun, radiant reaction of nuclear fusion that it is, projects a corona of burning gas and plasma out into the space surrounding it at all times. Ordinarily, we can't see any of this corona because it is hidden in the blue sky, lost in the glare of the bright sphere that generates it. During a solar eclipse, however, the moon occults the sun itself, leaving the corona clearly visible around it in the darkened sky. Now, because the sun is rotating, gas tends to jet out around its circumference more than it does at the axis of rotation due to centripetal force. Upon a solar eclipse, this lighter part of the corona also is visible, as wide, vivid flares to either side of the black disk of the moon -- as the wings of the falcon sun, gliding still and graceful across the heavens.
Egyptian myth backs this interpretation up, too, if you allow for a bit of Stephensonesque conjecture as you read it. At one point early on in the gods' lives, Osiris was god of the land and vegetation. Seth, god of darkness balancing light, god of change, and in this case god of evil, murdered him, and spread his parts throughout Egypt. Some time later, the parts of Osiris's body were reunited and burned, bringing his spirit back into the pantheon of the gods, now as the god of the underworld, and meanwhile impregnating Isis with Horus, aforementioned god of the sun. More time passes, and Horus finds himself in the position of avenging Osiris by doing battle with Seth. He calls upon the moon god Thoth, whose magic turns Horus into a special sun with outstretched wings and the goddesses Nekhbet and Uazet transformed into uraeus at his side -- or in other words, the moon eclipses the sun, showing its corona and diamond ring effect sparkles at its sides. Seth, darkness, is defeated by Horus, the sun, just as by definition some daylight remains after a solar eclipse.
Interestingly, this triumphant winged sun appears other in other cultures that took part in nascent civilization. The Assyrians portrayed their main deity, the sun god Asshur, as an eagle with spread wings and drawn bow. The Babylonians had Marduk, again the sun god, again the slayer of chaos as represented by Tiamat, and again an archer in a sun with outstretched wings. Even the Hebrew God, Lord of Hosts, is sometimes spoken of with wings, as in Ruth 2:12, Psalm 17:8, and the compelling Malachi 4:2, which begins "But unto you that fear my name shall the Sun of righteousness arise with healing in his wings ..."