How can underdeveloped societies grow economically and technologically without causing environmental harm to their own country as well as to other countries?
The only real way to do this is through technology that is environmentally cleaner than the typical technology available today. At today's levels of technology, most economic development is hard on the environment. Factories use up a great deal of energy that must typically be produced by burning fossil fuels, for example. Therefore, in order to experience economic and technological growth, most countries must degrade the environment to some degree.
The only answer, then, is to use cleaner technologies. It might make sense for these less developed countries to seek help from richer countries so that they might be able to build a clean infrastructure from the ground up. The less developed nations need clean technology that is likely to be beyond their means. Therefore, they are likely to need help from rich nations.
The only way to achieve the goals set out in this question is through clean technology. The best way for poorer countries to get such technology is through help from richer countries.