As someone who has studied archeology, this pertains to the Neolithic period which wasn't my specialty (my specialty was rather the hunter-gatherer societies of the Paleolithic). Dogs were first domesticated from wolves around 25,000 BC, when we were still hunter gatherers, because they were useful to defend our early tribes from predators. Bovines (cows) and ovines (sheep, goats..) were domesticated when we started being sedentary societies. That's around 12,000 BC in the near east, and 6,000 BC in Europe.

Anyway, we actually don't really know for sure how we figured we could drink cow milk, since by the time we domesticated bovines and most of our animals, we didn't have historical records. But we can take educated guesses.

In Europe and the Middle East, cows are the domesticated form of the Aurochs, which are now extinct (rather recently, surprisingly). The first sedentary societies likely domesticated the aurochs in order to firstly have fresh meat to eat, and readily available. But these early societies wouldn't let things go to waste, and thankfully our evolutionary history made us omnivores and able to consume pretty much anything. They naturally figured that bovine milk was also an available food source, so we used it. Basically, it pretty much came naturally.

Domestication results in many unintended and intended changes in anatomy to the animals. Among intended changes, which were consciously decided through selective breeding, are more "meaty" animals. That's grossly why a wild boar has a lot of mass in its front and head, whereas a pig (which evolved from wild boar through selective breeding) is fat in the backside, where there's more edible meat. Unintended changes due to domestication can be the color of the fur of the domesticated animals. That's why we have black and white cows, something which doesn't occur in wild bovine, but managed to be expressed during the domestication process because there was no natural selection to stop black and white cows from happening. Also selecting "friendly bovines" implies selecting those with "friendly genes", and those genes could be associated to different fur or skin colors (that's a work in progress... notably that's observed in the recent domestication of foxes in Russia, which unexpectedly resulted in unintentional differing fur colors although the scientists only selected for "friendliness").

Anyway, through selective breeding, as we liked bovine milk, we selected over millenia the bovines which produced more milk. Hence now a cow couldn't survive in the wild, because it needs to be milked constantly, more than its offspring could drink. Similarly, hens produce eggs regularly even though their eggs aren't fertilized by roosters. Because through selective breeding, we selected those cows, and hens, and pigs, and goats, who produced more milk, or more eggs, or more offspring, or more fat, for our consumption, until we created domesticated breeds which couldn't possibly survive without us. These animals were bred for our consumption, so naturally, we didn't want to waste anything, and we surely got around to using them in every way possible, be it their meat, their leather, their horns, their sheer usefulness to defend us or make things easier for us (for dogs, cats and horses notably), and their milk.

This is probably more of an /r/askscience question, since it likely started before recorded history in many cases. Cow's milk, which contains a lot more lactose than human breastmilk, is not digestible by most humans, and the numbers are heavily biased based on geography. Lactase persistence (the ability to metabolize lactose as an adult) is found in 80% of Europeans and many South Asians, but it is almost absent in sub-Saharan Africa and China, where people are nearly all lactose intolerant. This paper from 2006 studies lactase persistence in a subset of East Africans (Tanzanians, Kenyans and Sudanese) and found a set of genetic variants that are associated with the trait and estimate that this locus in the genome experienced strong selective pressure starting about 7,000 years ago. Since this is a separate genetic origin from the European event, we can't say anything about other groups, but we can at least make a reasonable claim about that particular set of Africans.

As I posted in another comment, there was a genetic study done in 2012 on cattle bones from an Iranian archaeological site dated to shortly after plant domestication began there. It found that all cattle living today are descended from as few as 80 animals that were domesticated from wild ox (aurochs) in the Near East some 10,500 years ago. This study was carried out by an international team of scientists from the CNRS and National Museum of Natural History in France, the University of Mainz in Germany, and UCL in the UK. A quote from one of the authors: “A small number of cattle progenitors is consistent with the restricted area for which archaeologists have evidence for early cattle domestication ca. 10,500 years ago. This restricted area could be explained by the fact that cattle breeding, contrary to, for example, goat herding, would have been very difficult for mobile societies, and that only some of them were actually sedentary at that time in the Near East.”

As for how people started drinking cow's milk, it would have begun during and after the time that people had access to tamed cows. Presumably it started in the Near East at about the same time that plant domestication began to occur. Both were not "revolutionary," but rather very gradual transitions that occurred over the course of at least a thousand or more years with generations and generations of genetic manipulation via breeding as well as restructuring of human subsistence practices, society and mobility patterns. It is thought that the process of willful plant domestication started before animal domestication. However, at some point people began to herd wild groups of animals. In the case of cows, these were wild ox species known as "aurochs."

Eventually these herded animals, over many generations, became genetically/morphologically distinct enough and behaviorally altered enough that they were almost completely dependent on humans for reproductive success. At this point humans were/are able to use them as draft animals in farming. They are also a stable, predictable food source in terms of the products they make. Milk is a really useful product because it can be fermented and preserved for long periods of time. In general, it is a steady source of calories for a group of people whose farming lifestyle is unpredictable due to weather change and associated famine/bad harvests/crop loss. There is a very large body of osteological evidence from many archaeological sites across the world--especially in early agricultural communities-- indicating that famine, child mortality, vitamin deficiency, and general malnutrition were common in these populations. Drinking milk and using milk products were one way to mitigate this and provide another supplementary protien/calorie source.

In Africa particularly, pastoralism (a cattle-based lifestyle in which people are migratory and not tied down by agriculture) has since at least 7,000 BP been a very successful life strategy for these same reasons of predictability as a food source and because it allows for constant mobility toward areas where plant resources are available. It still persists today as a very successful life strategy in "unpredictable" (food-wise) places such as the Kalahari or Namibian deserts. Given that much of Africa has become desertified in the last 10,000+ years, it isn't surprising that pastoralism is much more popular and successful in Africa than sedentary agriculture--though that isn't to say that sedentary agriculture doesn't exist. It's simply a less viable option in many parts of Africa.