This quote comes from Emerson's essay "Self-Reliance." In this essay, originally a speech given to college graduates, Emerson urges young people on the cusp of adult life to look inward rather than outward to find their God-given paths. They should, Emerson asserts, rely on their own soul's guidance over tradition or convention or what their parents want.
While it is best for one to examine one's own soul to find one's life path, Emerson also warns that this way of life is "arduous," or hard. He says in the above quote that two paths are easier than the one he advises. It is easy, first, to do what the world deems valuable ("live after the world's opinion") and also easy to follow one's unusual destiny in solitude, but the "great man" follows his unique destiny amid the conventional people who will criticize and judge him.
This quote is important because it points to the reason behind relying on the "iron string" of trusting one's own inner voice. Emerson preaches nonconformity not because it is a selfish way of indulging oneself, but because it serves the greater good of humanity. You don't follow this path for yourself, but for others. He mentions a list of nonconformists who changed the world for the better, such as Jesus, Socrates, Galileo, and Newton. They made the greatest possible impact on society because they refused to go off in solitude. Instead, they lived in the midst of society and accepted the suffering that path brought.