The driving force or primary motivation of Emerson's belief is that individuals think for themselves and not seek to follow a conformist social setting. This is expressed in many points in his writing: "Trust thyself: every heart vibrates to that iron string." This is something that we teach children from the earliest of ages in our schooling. We no longer teach kids that we come from a "melting pot" where everyone is homogenized and is the same. Rather, we instruct students as to their unique differences and we teach in curricular settings that are meant to laud and praise such level of difference: "Nothing is last scared by the integrity of your own mind." The essence of modern character education and development in our public schooling is to drive home the idea of difference and diversity, the main idea of Emerson's writing.
It is hard to say for sure than anyone in today's world has been influenced by Emerson, but you can clearly see things today that look like they might be influenced by the sort of ideas he put forth in "Self-Reliance."
To me, the United States today is a society where so many people think that it is okay to march to their own drummer. In the old days, people seemed much more concerned with how society would see them. They did not dye their hair funny colors or get piercings in "weird" places. They did not get tattoos or engage in the kind of "look at me" celebrations that you see athletes doing from the pros down to high school.
You can argue that our society is so permissive today because of ideas like Emerson's. You can argue that his ideas make us all think we should do what we want to do without paying attention to what others think of us. To some extent, this is good, but I think we have gone a little too far and could use some more attention to common values and a little less emphasis on each person being their own person.