The meaning of this question is not completely clear. It could mean two things.
It could mean “why do nations like the US want to influence developing countries to have democratic governments and free markets?” In this case, the most likely answer is that they want to have countries develop these things because they believe that these systems are the best way to organize societies. Practically every developed country in the world has democracy and a free market to a fairly large degree. Therefore, these institutions seem like the best way to political stability and economic growth.
The question could also mean “why do the developing nations allow the rich world to influence them?” In this case, there are two possibilities. One is that they might actually believe that democracy and capitalism are superior systems. They might see the success of countries with those systems and want to emulate them. On the other hand, they might simply be allowing themselves to be influenced as a way to ingratiate themselves with the rich countries. They might think that the rich countries are more likely to give them aid if they move towards democracy and free markets.