What ties together their poster boys of South Korea, Taiwan and Singapore? They all had a baseline of education and decent government (for corporations, anyway), which they used to attract foreign investment (read: money flowing into the country) for industries that employed large amounts of people in roles where they could learn useful skills, kick-starting middle class wealth that could then propel the rest of the economy forward.
Most poor countries only have investment in extractive industries (oil, minerals, etc), which employ very few people, and the skills they learn are next to useless for anything else. The next rug up the ladder is places like Cambodia and Bangladesh, where investment goes into labour-intensive but low-tech industries like making clothes, but the government is too corrupt and the people too badly educated for much more. Although Bangladesh, in particular, has been a quiet success story lately, eclipsing India in GDP per capita.