I'm going to give you a name that you probably didn't expect when you asked this question:

John Snow

No, not everyone's favorite valyrian steel wielding bastard, but the so called father of modern epidemiology. When you read through that biography it gets you a sense for the lack of understanding of epidemiology up until the mid 1850s. Until that point people thought disease was carried through other supposed wrongs - like air pollution. I mean think about it, the idea that there are tiny microscopic critters swimming around in everything that live and die without our knowledge is a rather revolutionary concept. It took a particularly deadly waterbourne disease, cholera, and it's spread to Victorian England to really provide the impetus to study this field.

As far as to why people didn't seem to notice that boiling water made it "healthier" that's difficult to answer. As a historian, you can't really prove why something didnt happen, you can only study and describe how it did. But in modern contexts, we tend to boil untreated water when we are ingesting microbes that we are not used to. For example, when a water treatment plant fails, or if we are in a locale where our bodies are not used to the particular biota that lives there. You can see how this probably wasn't a huge deal for people living 1000s of years ago. If you lived in an area and consumed a certain water source your whole life, you would rarely get sick off that source unless an outside pathogen was introduced- which was more uncommon in days of yore. Thus, when cholera was introduced, a particularly severe pathogen, it caused people to take notice. Without that forcing function, the innovation of pre-treating water was not as obvious.

I hope this helps answer your question.

I suppose that an important part of your question is to recognize that even within the our current medical understanding, the statement "boiling water prevented disease" is not universally true. It may seem kinda trivial to point out that things like TB, cancer, and the flu can't be prevented by boiling water. But if what you're trying to say is that the fact that Cholera and some other gastrointestinal infections can be prevented by boiling, thus why did people not boil their water? The answer has multiple parts, a lot of them centered around the point "how could they know?" Here are a few of them:

First, and I think that this is a big one, prior to the 19th century, Cholera (the epidemic gastrointestinal disease) was not really a problem until the late early modern period. Charles E. Rosenberg points out that prior to 1817, "there had probably never been a cholera epidemic outside the far east" (1962:1). So I think it is relatively safe to say that the signature disease that boiling water can prevent wasn't even a concern for the western medical community until the 19th century.

Second, the drive to prevent diseases before they start---at least in the fashion that we now work with an eye towards etiological agents---has not always fit in the realm of actual medical practice. Medical professionals throughout much of Europe's history have been primarily concerned with treating diseases that were already in course. The questions they were often asking was how do we bring the body back to health. Diseases, in their many forms, seemed (and still are) a normal part of everyday life. The idea that states and professional organizations can intervene at the population level to transform everyday life, and thus reduce disease, is a modern idea. Michel Foucault is one of the key thinkers here that pointed out those specialists concerned with demographics were engaged in a new thing---``noso-politics,'' where the widespread illness was recognized as both falling along class lines and also have economic repercussions (1972).

And, thirdly, this new concern with demographics of health and disease created a whole set of institutional motivations that would bring about new ways of thinking about disease. As just one example, Andrea A. Rustnock writes about how in Eighteenth-Century England, medical practitioners and bureaucrats had to rethink the nature of disease and the role of medicine. Industrialization and commercial imperialism was moving people around the world and closer to urban centers. And intellectuals and politicians were becoming concerned with things like epidemic disease and health insurance. This changed the practice of diagnosing cause of death from being a local concern to a now national-scientific concerns. Prior to new noso-political interests, English physicians were not to concerned with determining cause of death from illness. That was a job done by a group of people called Searchers---``who typically were older women paid by the parish to examine corpses and ascertain causes of death'' (2002:19). But whether searchers were medically educated or not, the process of diagnosis was both political and prone to local idiosyncrasies. A highly admired and esteemed community leader dies of some wasting disease? Well, you (the searcher) probably doubt that he was depraved enough to contract syphilis or consumption, right? Now it wasn't simply enough knowing a cause of death, there was a political motivation to rationalize and standardize the classification of death.

I hope that this then gives a sense to the complexities of medical knowledge (both then and now). You see, before we can make the statement "boiling water prevented disease" sensible, we need to recognize categories of disease are somehow substantive. If causes of death (just to name a few discribed in a 1750 London table of common deaths) could be colds, head-ach, mortification, planet-struck, [just] spleen, or lunatic---then you might have a hard time understanding how heating water to a boil can prevent any disease or death. But now we believe that there are a specific class of disease causing organisms that enter the body through the mouth through contaminated drinking water that can be prevented by killing these micro-organisms with heat. But such a classification of diseases did not exist, and was not widely accepted until the late nineteenth century.

And the question of how we got to that point is actually a really big one that is still studied by historians (it's my area of study actually). But there are a lot of parts to it. For sure one of the major elements is that the nineteenth century hygenic movements---involving everything from municipal trash collection to the science of indoor plumbing. Tomes' The Gospel of Germs is a really good account of turn of the century American public health work.

But perhaps the most important way to think about this question is instead of asking "why no one figured it out," ask instead, "what does it take to get people to boil water before they drink it"? Because the second question is much more interesting, and it's open to a more careful kind of investigation.

Rosenberg, 1962, The Cholera Years, University of Chicago Press.

Foucault, 1980, Power/Knowledge: Selected Interviews and Other Writings, 1972-1977 Pantheon

Rusnock, 2002, Vital Accounts, Cambridge University Press

Tomes, 1998, The Gospel of Germs, Harvard University Press

No one knew what caused disease, since germ theory hadn't been formed. Therefore, the connection between killing bacteria by boiling water and preventing disease wasn't known.

However, it was certainly not unknown that boiling water reduces the risk of disease. For example, both Paulus Aeginata and Hildegard von Bingen, among many others, wrote about the beneficial aspects of boiling water. There are earlier Jewish, Greek and Sanscrit texts discussing boiling water.

For most of Western history, however, from Galen to Pasteur, people have been more concerned with the effect of water upon body fluids and the spread of foul air from decomposing matter. As water was cold (even if boiled) it was thought to produce excess phlegm and consequently disease.

Followup question: It is the norm in Asia to boil all drinking water, all the time, even in places like Taiwan which now have sufficient infrastructure to prevent disease. Does this custom predate germ theory or is this a recent phenomenon? Is the correlation between drinking unsafe water and getting sick for some reason more obvious in ancient China?

I hope this isn't off topic, but a Roman teacher named Marcus Terentius Varro speculated that "tiny creatures flew into our noses and mouths" and this is what caused disease. While a bare bones theory compared to what John Snow achieved, why didn't the idea at least gain some traction in the medical community, considering he wrote this idea a while ago?

Related question:

If cowherds knew they were immune to smallpox why dint anyone believe them and then start hanging around cows more often?

All these people have given you a great deal of information about the beginning of germ theory, but to that I will add another interesting part of medical history. Hydropathy was a system of medicine developed in the early 1800s. The idea was that one could cure themselves of physical ills by the application of water cures externally and internally. They recommended teas and drinks that would require the boiling of water, so in that way this type of medicine probably did, at least, not harm their patients. Some of their other cures were dubious, but probably promoted cleanliness.