The myths themselves are not going to give a technical explanation of this, so we are setting off on the wrong foot if we are seeking an answer at such a level, which is that of our modern technological mind.
An initial observation to make is that the myth does not always include a shield, as Yannis notes in the case of Hesiod's account. We can also turn to the iconography, which likewise also does not necessarily portray a shield. See, e.g., the metope from Salinas ca. 540 B.C. (at the Museo Nazionale Archeologico in Palermo, Italy; in Morford & Lenardon's Classical Mythology 8th ed., p. 552) and the black-figure jug by Amasis, Attic, 6th cen. BCE (in Neumann, Consciousness (below), illustration 25) both showing Perseus killing her while simply averting his gaze. For the Greeks, averting the gaze and looking at her reflection in the shield/mirror were obviously equivalent. Her power was either nullified or greatly diminished when looked at in reflection.
I also don't think the eyes were crucial. I'm not aware of any source that stresses her eyes. Rather, they focus on her whole face (Ovid, Metamorphosis 4:783) or overall appearance, as in the passage from Apollodoris in the answer above. We can't ignore her protruding tongue, tusks, or snakes for hair, since they are all essential to her character and mythological meaning. That's why they are there. And that's why Perseus had to look at the shield even though she was sleeping (presumably her eyes were closed).
If we go into the underlying psychological aspects of the myth (as we should with most myths), Perseus, like the typical hero, is in the process of fully realizing his ego consciousness, while this monster represents the all-consuming Terrible Mother of the unconscious seeking to suck him back into there and destroy his nascent ego consciousness. To confront her directly is too dangerous, so only averting the gaze or looking at her indirectly through the reflection will enable him to succeed. In the versions where the "bronze" shield/mirror appears, that is a solar/light symbol of the ego that not only enables him to see a diminished Medusa, but which as such also serves as a guide and beacon to him to keep him on course. Here it is also relevant that the Greeks also stressed the details of his escape. When Perseus kills the Medusa, the winged horse Perseus appears out of her (representing the ego that has escaped her, a parallel with Perseus himself), which Perseus rides while being pursued by Medusa's sisters. His winged sandals, invisibility helmet, and hiding-wallet also play a figurative role, for which he has to thank Athena, goddess of wisdom (and so of consciousness). See Edward Edinger, The Eternal Drama: The Inner Meaning of Greek Mythology, pp. 4, 83; Erich Neumann, The Great Mother, p. 166; Erich Neumann, The Origins and History of Consciousness, pp. 87, 178, 214-16.
I hope this helps.