I've just always wondered what factors have allowed for the explosive population in India, and why it occurred there as opposed to any other number of locations.
initial conditions.
India (and China) both have large, perennial river systems that irrigate a massive fertile plain. they have been settled continuously by complex civilizations that were relatively technologically advanced (with the wheel, metallurgy, arrows, domestication and multiple food crops) for thousands of years. they both got a head start in clearing out the large forests[1] that covered their lands (again, due to their tech.).
remember that the biggest rivers in asia: the indus, the ganga, the brahmaputra and the rivers in china are all glacier fed and come from the himalayas. they are reliable, don't flood that much, and carry massive amounts of water and silt.
several thousand years of reliable agriculture will tend to cause your population to grow exponentially.
in contrast, other ancient river systems that were technologically advanced (egypt, the fertile crescent) were comparatively much smaller, and quickly reached carrying capacity. [2]
2000 years ago, almost the whole of the indian subcontinent was under the rule of a central empire, that was fully capable of harnessing the massive agricultural potential of the indo-gangetic river system. in contrast, europe lacked the pleasant weather, the abundance of fertile lands, and the social and technical tools required for large, centralized food production.
as for the new world, the absence of, or total integration of critical technologies like metal working, domestication of cattle and the lack of intensive farming of crops like rice meant your growth was always stunted by a constrained food supply.
jared diamond's guns germs and steel, (also "collapse") fleshes out this idea a bit more.
[1] i don't know about china, but almost all of india was supposedly covered by a dense impenetrable forest called the "dandakaranya" that might have been bigger than the amazon. it's all gone now, presumably because immigrants with metallurgy and domesticated animals moved into india ~3000+ years ago and razed it down.
[2] in comparison to the indo-gangetic plain, the fertile bits of egypt are tiny. and surrounded by lots of desert. the same is true for the fertile crescent in modern day iraq. in india, however, even the bits that aren't in the indo-gangetic river system are quite fertile, with large seasonal rivers and sometimes two seasons of rain.
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Traditionally, people would have massive families. Under British rule, most of the population was relegated to rural activity so having larger families meant more workers on the farm. Couple that with high death and IM rates and people had as many children as possible.
Another reason is food production. India is incredibly vast and warm. Today, the country has a net import of food of zero meaning that they grow everything they eat. If the country can eat just local food, there is no cap on population. Also, the green revolution greatly increased the output of food production, leading to a greater population.
Lastly, India's rapid modernisation (including the green revolution) resulted in a crash in death rates (relative to the population). People today live so much longer yet at the same time most people still have the same number of children. Better food, better medical care, and more knowledge of hygiene has created an environment where the crude death rate has remained steady while the crude birth rate rises.
Basically, due to the continuation of high birth rates through the '60s-90s the population exploded (NC over 20 for all those years). Combine that with increased access to good food via the green revolution plus access to good healthcare and voila, a massive population boom that has lasted until today. Today, due to more urbanisation and a wealthier populace, the birth rates have fallen in cities while remaining steady in rural areas.
Final point: This chart indicates that between 1900 and the '30s the population remained relatively stable. Also, at independence (1947) the population was about 350 million. It wasn't until the green revolution that the population began to increase by millions a year.