This is Saturn.
In every picture of Saturn, we can see that her rings are arranged in a perfect 2-d plane.
So the question I ask is simple, why are they arranged so? Why aren't the rings arranged in the 3-d space around Saturn?
I know that the orbits of planets around sun is also 3 dimensional, then why doesn't it apply to the rings of Saturn as well.
I think your question is a matter of scale. Various astrophysical objects, ranging from spiral galaxies, solar systems, to the rings of Saturn form as a result of an initial spinning ball of dust and gas. All three of these objects are on very different physical scales, but each one flattens out with time into a disk. The solar system is approximately a disk, our galaxy is a disk, and so are Saturn's rings. The only difference between the three in the vertical extent of the disks--our galaxy is a lot thicker than Saturn's rings, but it also has a much, much larger radius.
Any object spinning about an axis--even the earth--will flatten slightly into an oblate spheroid. If the spinning object isn't held together strongly, its component particles--whether they be rocks in Saturn's rings or stars in our galaxy--will get pulled outward by centripetal forces and eventually turn the system into a disk.
The same physics is at play at all scales.
