This image shows that the rear side of Saturn rings is dark.
These images show the variation of the rings' brightness: the closer Saturn is to Earth, the brighter its rings appear.
The rings of Saturn consist of 99% of water and the remainder are various impurities.
The water is in the form of water ice. Since ice is a crystalline structure, it reflects and refracts light, making the rings appear bright.
But the brightness also depends upon the intensity of incident light: the closer Saturn is to the Sun the bright its rings.
The reason the rings of Jupiter appear dull (even though it is on average closer to Earth than Saturn is) is that they are composed of dust, which is darker than ice.