Let's take a closer look at the full quotation from Emerson:

Trust thyself: every heart vibrates to that iron string

Emerson explains fairly explicitly what he means by this comment. In writing on the virtues of individualism and self-reliance—virtues which had been championed by various previous poetic movements, including Romanticism—Emerson is stipulating that we must trust ourselves because each person's heart "vibrates" according to one's own values and determination. The "power" within each person, a power which sets the string of one's heart trembling, is completely different to any power that has previously existed in any other person. Emerson uses musical imagery to underscore this fact: just as a string will vibrate to a different pitch depending on how loosely or tightly it is tuned, and will sound differently depending on the "power" with which it is struck, each person makes a different impact upon the world and is guided by one's own particular sense of self.

The quote seeks to embrace much of the Transcendentalist philosophy of which Emerson was a part.  This movement sought to develop knowledge and understanding within the individual.  At a time in American History, when industry and commerce was taking a strong hold in America and individuals acted in concert with one another in an almost conformist manner to make money, thinkers like Emerson sought to bring back a sense of individual voice into this discussion.  For Emerson, "Transcendentalism revolves around the self, specifically the betterment of the self."  This was outside of the realm of religion or social acceptance.  The quote bring this idea out.  In it, "that string" is the development of individual identity and hearing one's "voice."  This is a pursuit that is individual, bringing out the idea that each person holds a certain uniqueness that is different from all others.