This is actually a pretty good question.

The answer is that before industrialization in the early 1800s, Japan was one of the richest countries in the world, on par with Britain, Northeastern United States, Yangtze Delta China, and Bengal India. You can read a short version of this in Susan Hanley, “A High Standard of Living in Nineteenth-Century Japan: Fact or Fantasy,” Journal of Economic History 43 (March 1983): 183-192. Hanley directly compares Edo (today's Tokyo) with London and concludes that if she were a commoner, she would choose Edo and if she were a member of the elite she would choose London.

So the high level of economic activity before industrialization led to a relatively high population before industrialization. The extreme poverty you mention is a myth (which was busted in the 1980s).

For China, I recommend R. Bin Wong, China Transformed: Historical Change and the Limits of European Experience (Ithaca: Cornell UP, 1997) and Kenneth Pomeranz, The Great Divergence: China, Europe, and the Making of the Modern World Economy (Princeton: Princeton UP, 2000). For India I recommend Prasannan Parthasarathi, Why Europe Grew Rich and Asia Did Not: Global Economic Divergence, 1600-1850 (Cambridge: Cambridge UP, 2011). Timur Kuran, The Long Divergence: How Islamic Law Held Back the Middle East (Princeton: Princeton UP, 2011) discusses the Middle East and shows how the 18th century was when the Middle East began to lag.

Now why does Asia have a higher population density in general today? The answer to that question is rice. Rice produces more calories per acre than wheat (about 1.5X, rice paddy 1482kcal/sq m: wheat 1083kcal/sq m).

So compared to wheat growing regions, rice agriculture can sustain more people. You can check the right most column of this webpage for the cal per sq meter.

http://www.gardeningplaces.com/articles/nutrition-per-hectare1.htm