What chemicals cause the softer, less-raw taste of purified water sold in bottles?
Water issues tend to be very specific to a certain location, so it is not possible to say exactly for sure what is happening in your situation.

Pure water has no taste.  The taste comes from the impurities.  Impurities, which are almost always present, can partially be removed by the filter shown.  That filter looks like a carbon filter which is effective on chlorine, organic particles, and sediment.  This filter is ineffective on minerals and inorganic compounds.  

Minerals give tap water its hard taste. This hard taste is harmless (and may actually serve as a dietary mineral supplement.) Removing the hard taste requires a process like deionization or distillation which is costly. The cost of the water can go up significantly with more mineral removal.

Chlorine, which is used to kill pathogens, can be neutralized with citric acid.  Squeezing a lemon or lime in the water can cut down on a chlorine taste.  

Hard water can be chemically softened with something like baking soda; however, this is generally trading one ion for another, so the water will only taste softer. 

Try adding a reverse-osmosis filter after the filter shown. The filter shown will treat the chlorine while the reverse-osmosis filter can treat the minerals. (Chlorine may damage a reverse osmosis filter, so it is best to get it out first.)