On the broadest level, the average density of Saturn, or any other planet, is a function of composition, pressure due to gravity, and temperature.
Unfortunately, only the composition of Saturn's atmosphere is directly measurable, while the composition of the planet as a whole is speculative to some degree.
The Voyager mission initially measured the atmospheric composition of Saturn to be 96 mole percent hydrogen and 3% helium, or 94% hydrogen and 6% helium by mass (see Conrath et al. http://adsabs.harvard.edu/full/1984ApJ...282..807C ) This is a much higher percentage of hydrogen than any other planet. Subsequent research and reanalysis of data has increased the accepted value of the ratio of helium to hydrogen, but it is still accepted that Saturn has a greater atmospheric precentage of hydrogen than any other planet, and that hydrogen and helium account for >99% of the atmosphere.
On the other hand, it is thought that that the measured composition of Saturn is not reflective of the interior composition, because liquid helium may rain down to a lower level on Saturn, and to a greater degree than Jupiter due to cooler tempeature.
It is thought that Saturn has a rock/metal core of no more than 22 Earth masses, but no lower limit can be placed upon the size of the core (see Guillot http://adsabs.harvard.edu/abs/1999P%26SS...47.1183G).
So Saturn should be thought of as a small dense core surrounded by mostly hydrogen and some helium.
For both Jupiter and Saturn, it is believed that just outside the core is a large mantle of metallic hydrogen. However, because of the greater gravitational pressure of Jupiter, the metallic hydrogen layer is considered to be substantially larger in Jupiter.
The density of metallic hydrogen is 0.6 grams/mL at 140 GPa and 3000K ( https://e-reports-ext.llnl.gov/pdf/244531.pdf ), of course varying with temperature and pressure.
So think of Saturn as having a small dense core, surrounded by a layer of metallic hydrogen, and an outermost layer of mostly ordinary hydrogen and some helium of less density than the metallic hydrogen.
When thinking in comparsion to Jupiter, consider that even if the measured higher percentage of hydrogen on Saturn is not reflective of the interior, greater compression of the hydrogen and helium within Jupiter causes a greater density, especially due to a thicker layer of metallic hydrogen within Jupiter.