Well, there is hard science behind the fact that water comes in various degrees of hardness/softness depending on how much Calcium and Magnesium are in the water.
In the same way that Dripstone caves form, hard water can (and will) leave deposits of calcium and magnesium compounds in your appliances (Limescale). From that article:
Calcium bicarbonate is soluble in water, however at temperatures above 70 °C (158 °F) the soluble bicarbonate is converted to poorly-soluble carbonate, leading to deposits in places where water is heated.1 Local boiling “hot spots” can also occur when water is heated, resulting in the concentration and deposition of salts from the water.
Calcium cations from hard water can also combine with soap, which would normally dissolve in soft water. This combination often forms scum which precipitates out in a thin film on the interior surfaces of baths, sinks, and drainage pipes. Soap usually contains salts of anions from neutralized fatty acids or similar chemical compounds. The calcium salts of these anions are less soluble in water.
So up to now, everything is all nice and scientific. Next: Water softeners. There is a chemical process that will replace the calcium and magnesium ions in your water with sodium ions. The resulting compounds are much more soluble in water and do not form the unsoluble compounds, so no limescale will form. Again, this is hard science: You let the water pass through a filtering system that contains certain molecules with sodium attached, whose chemical properties allow the sodium ions to be easily replaced by the calcium and/or magnesium ions. After a while, the filter becomes saturated with the Ca and Mg ions and has to be regenerated; again, there are chemical processes that will accomplish this.
Next: The effects. It will prevent your pipes and appliances from being clogged by limescale. But what about drinking the softened water? Hard water is not unhealthy, and softened water is not healthier. In fact, since it contains more sodium than normal water, you should be careful if you have to follow a salt-restricted diet. Also, calcium is important for bone stability. For this reason, many people opt to have one tap bypassing the water softener.