Got this from Wikipedia.
link
I think it’s interesting to look at this graph in terms of migration and colonization. Northwestern Europe is VERY lactose-tolerant, and most white Americans are descended from NWE stock, so there’s plenty of lactose-tolerance here (with African-descended and Native Americans the exception). Likewise with the lactose-tolerant whites in Australia (from NWE), and the lactose-intolerant Aboriginal Australians.
Then you have that belt of southern France, Italy, and the Greek peninsula where lactose tolerance drops; continuing that trend westward, it would make sense that Spain and Portugal would have low tolerance as well. Portugal and Spain had a strong influence in the European infiltration of South America, and as the indigenous South Americans have little lactose-tolerance (much like their North American counterparts) you still have an overall low incidence of it in South America today.
And in Asia, where lactose-tolerance is VERY low, you have a gray zone (no data) in Mongolia, but considering that the Mongolian diet is rich in yak milk, yak butter, and so on, I suspect their lactose-tolerance is fairly high. Then you have India, where there is a spike in lactose-tolerance, perhaps because of the reliance on milk, which is the basis of the Hindu reverence for cows?
I’m curious about the sharp divide between the southern African countries and the central ones. Ancient tribal lineage? Herding culture vs. non-herding? Effects of the US-colonial-era slave trade and mixing with westerners?
Sean K Reynolds 
The Mongolian thing, I wonder whether the composition of yak/reindeer/horse milk is different enough from cow milk that it’s easier to digest for people who are lactose intolerant/mildly lactose intolerant? Sort of like the way goat milk and cheese is a little easier to handle (maybe that’s part of why southern France shows up the way it does – a fair amount of milk, but not neccessarily cows).
Reading up on LI on wikipedia, horse milk has more lactose than cow’s milk, and goat milk has lesslactose than cow’s milk. So the Mongolian reliance on horse milk works nicely with their higher tolerance. And good eye on the France/goat cheese thing, which also explains goat’s milk in Greece (frex, feta cheese).
My own opinion is that the primary factor is simply a difference in culture. Some chose to drink milk, others didn’t. :)
I recall reading somewhere (wish I could recall where; this was years ago) about the early Viking expeditions to the New World, and how relations turned sour when the Vikings shared their milk with the native Americans. The native Americans, not being milk drinkers, got sick and very angry, thinking the Vikings had tried to poison them.
The Vikings quickly decided they should be somewhere else. ;)
Except that LI is genetically-based. It’s not a matter of Native Americans not being milk-drinkers, they actually lack the gene that lets them digest milk as adults (we all can do it as infants, obviously). The graph might as well be a graph of the dispersal (or lack thereof) of the gene.
Hmm…
Definitely a puzzler then. If I were a scientist, I might have to begin a proper study on it.
I’m slightly lactose intolerant. Strangely, I can drink more than two cups of cold milk and have only a mild reaction, but a half cup of warm/hot milk and I am wracked with pain and cramps. Unfortunately this means Starbucks’ chai tea latte is off-limits, which makes me very sad.
Milk is gross though, so I generally don’t miss it.
The trends in Africa
I really don’t know if this is significant at all, and it seems it shouldn’t be, but the area marked as having low incidence of lactose intolerance in Africa maps fairly well to French colonial possessions.