What exactly does carbonated water do?

Carbonation produces the textural effects associated with effervescence, first of all, and a lot of people find that texture enjoyable. Texture is a huge part of the culinary experience and affects how flavorful compounds contact your tongue, thus affecting its perceived flavor.

Carbonation also raises the acidity of a beverage slightly, due to the presence of dissolved carbonic acid in the solution. As @Joe mentions in comments, this can promote the release of gasses with additional dissolved flavorful compounds. All of this can have a suppressing or enhancing effect on other flavors (a topic which is absurdly complicated and still being explored by food scientists much smarter than I).

Basically, the point of carbonating is similar to the point of pureeing something, or the point of dehydrating it. It's a fun and easy way to change how you experience the food--or beverage in this case.

I find sparkling water more refreshing than still water. I also find it to be a better palate cleanser. There are no side effects. Carbonation simply changes the experience, for me, making it more pleasant. There is also research that suggests carbonation changes the way the mouth perceives sweetness. Further, an article was published last year, in the journal Food and Function, with the title "Sensory attributes of soft drinks and their influence on consumers' preferences". So the point of carbonation is typically because consumers enjoy carbonated beverages. Are you a beer drinker? Do you prefer flat beer or carbonated? Wine drinker? Wouldn't you say sparkling wines and still wines are experienced differently?

See the top answer here. I think it covers this pretty well: "CO2 tastes good! Carbonated water is actually a weak carbonic acid solution, this is what you taste"