Whisk(e)y has some crazy chemistry going on inside of it, due to the complex interactions between water, alcohols, oils, esters and other compounds of various complexity. The profile of these chemicals will vary between different whiskey/whisky styles, but the overall chemistry is similar.
Simple effects of dilution
Adding water, or serving on the rocks, has a number of simple effects, such as diluting the ethanol a bit (ethanol anesthetizes your taste buds a bit), and cooling it (making your taste buds slightly less sensitive to certain flavors), but the fascinating part is what happens to the oily flavor compounds during dilution.
Dilution masks some flavor compounds
Oily hydrocarbons are somewhat soluble in high-proof whiskey. There are long-chain esters and short-chain esters of many varieties. As you add water, the whiskey becomes more polar, and the long-chain esters become supersaturated and start to precipitate in the form of micelles, microscopic "droplets" of esters that have clumped together. In some liquors like absinthe or ouzo, these droplets can get so large that they become visible, and visibly cloud the drink (an intended feature of absinthe preparation). In whiskey, these droplets are usually microscopic and don't visibly cloud the drink, because most of the oils have been removed during chill-filtration.
However, these droplets do something important, in that short-chain esters, being more soluble in the droplet than they are in the diluted whiskey, enter the droplet and become trapped inside. These compounds are now less available for tasting or smelling. Fortunately, these compounds are the oily, grassy compounds that many people do not like in their whiskey, and masking them is considered an improvement.
Dilution releases other flavor compounds
There is another type of micelle "droplet" that forms in whiskey. Ethanol, in high concentrations in water, forms it's own clusters, as ethanol molecules gather up with one another. Interestingly, warmer solutions cause more clustering of ethanol molecules, as do higher concentrations. Like before, these micelles trap compounds that are more soluble in ethanol than they are in water, volatile flavor compounds. However, unlike the oil droplets, these flavor compounds are desirable. Cooling the solution and diluting the solution both serve to "pop" these ethanol micelles, allowing them to release their trapped compounds for aroma and flavor.
So cooling and adding water can have the effect of both masking certain flavors by forcing them out of solution, and enhancing others by promoting their release back into solution. In the end, the result of the changed flavors is a matter of taste, which is why some people prefer neat, with water, or on the rocks, but one cannot deny that real chemical changes are in play.
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Best Answer
I too had never heard of this until I read @user5561's answer. So although it's news to me, I'm going to venture an answer. I believe the "reasoning" is as follows:
Now there are three assumptions this belief will be predicated on:
I take issue with the first two assumptions. You won't be able to tell the difference unless you're using water you just scooped out of a swamp and, with good water, there's no reason to believe it could not possibly improve the drink. I doubt the third but don't have any evidence either way.
I've come across a similar argument used for pizza, coffee, stock, etc... For example, since coffee is mostly water it's the most important ingredient. It seems logical until you realize that the neutral flavour of water is so subtle that it's quickly overwhelmed by just about anything you add to it (especially ground coffee beans).