When seen edge-on, Saturn's rings all but vanish, as this beautiful image, captured NASA's Cassini spacecraft and compiled by amateur astronomer Fernando Garcia Navarro, clearly illustrates.
Saturn's rings are about 175,000-miles across, but just 3,200-feet thick. Think about that for a moment. The ratio is mind-boggling. If you had a model of Saturn that was 3 feet wide, its rings would be roughly 10,000 times thinner than a razor blade.
The animation below shows the view from the Cassini spacecraft during what's known as a "ring plane crossing." Via NASA:
The movie begins with a view of the sunlit side of the rings. As the spacecraft speeds from south to north, the rings appear to tilt downward and collapse to a thin plane, and then open again to reveal the un-illuminated side of the ring plane, where sunlight filters through only dimly. The movie consists of 34 images taken over the course of 12 hours as Cassini pierced the ring plane.
NASA claims that six moons pass through the field of view during the sequence I've animated above. Maybe you'll have better luck with the original video, but I can only spot the two big ones, which are, in order of appearance, Enceladus and Mimas.