According to Astronomy Magazine, the moon turns red during a lunar eclipse because it is illuminated by the light that refracts through Earth's atmosphere. While the moon may pass into Earth's umbral shadow, the light that bends around the planet is enough to give its satellite an eerie red pallor.
Whenever one celestial body eclipses another, it can create two types of shadow. The penumbral shadow occurs when some of the sun's light is eclipsed, while the umbral shadow represents a complete blockage of the sun's light. For instance, when the moon eclipses the sun, anyone standing in the umbral shadow sees it as a complete eclipse with the sun completely covered by the moon's disc. Those in the penumbra are still able to see the sun's light around the moon's surface.
Since Earth has an atmosphere, however, light behaves differently when it eclipses the sun. The body of the planet may block all light from reaching the moon's surface directly, but light that passes into the atmosphere bounces and refracts off oxygen, carbon dioxide and water droplets in the atmosphere. To someone standing on the moon's surface, it would seem that Earth was surrounded by a bright halo of light, and this secondary illumination is enough to give the moon's surface a dark red coloration during a full eclipse.