The correct answer is to win the respect of other industrialized countries.
The White Man's Burden is a concept that establishes the idea that it is up to white, European countries to "civilize" other nations/territories that are not as developed. The goal of taking up the White Man's Burden is to help other nations become more industrialized and spread their religion (usually some form of Christianity).
In the aforementioned stanza, the last line talks about "the judgement of your peers." This clearly references the idea that industrialized nations need to take up this burden in order to be respected by other industrialized countries. This idea would help to increase imperialist ideas during the late 19th century.
