Light from the Sun is reflected off of the Earth onto the Moon. Some of this light is then reflected off of the Moon back towards the Earth.
Because of this we see part of the Moon illuminated by the Sun and the rest of the Moon dimly illuminated by this doubly reflected light (which we call earthshine).
Each time light reflects off a surface (like a planet), it gets dimmer because some of the light is absorbed by the reflecting surface.
This means that earthshine is dimmer than moonlight because earthshine is sunlight that has been reflected twice and moonlight is sunlight that has been reflected only once (off the surface of the moon). In addition, the reflectivity of the moon (its "albedo") is less than that of the Earth, which makes earthshine even dimmer.
Have you ever wondered why the moon is more orange or yellow in color when it first rises at night? This effect is caused by the atmosphere of the earth.
The reason for the orange color is due to the scattering of light by the atmosphere. When the moon is near the horizon, the moonlight must pass through much more atmosphere than when the moon is directly overhead.
By the time the moonlight reaches your eyes, the blue, green, and purple pieces of visible light have been scattered away by air molecules.
That's why you only see yellow, orange, or red.