While Emerson does claim that "Nothing is at last sacred but the integrity of your own mind," he doesn't make this claim until the sixth paragraph of "Self-reliance."  In the fourth paragraph, Emerson focuses on other, related concerns and uses them to build toward the idea.  Here Emerson focuses his attention on children, and the ways in which they more accurately represent more conscious human beings.  As THE Transcendentalist, Emerson regarded nature as the end-all, be all of goodness and purity.  For Emerson, God is in every aspect of nature.  In this paragraph Emerson suggests that the "behaviour of children, babes, and even brutes" are "pretty oracles [which] nature yields to us."  In other words, though the acts of the young and brutish, we see our true nature.  He continues:

Their mind being whole, their eye is as yet unconquered, and when we look in their faces, we are disconcerted. Infancy conforms to nobody: all conform to it, so that one babe commonly makes four or five out of the adults who prattle and play to it.

Unlike adults, who have been worn down and weather-beaten by the world, by society, children are the unbridled conduits of truth.  While often thought of as weak, immature, and undeveloped, Emerson states that youth is "sufficiently clear and emphatic."  Youth is not underdeveloped as it is thought to be; on the contrary, youth represents the highest level of truth and understanding of our true nature.  It is the acts of growing up and learning how to fit into society that actually constitute the regression of our true self.  This is why, two paragraphs later, Emerson states that "the integrity of your own mind" is the most sacred thing there is.  However, in this specific paragraph, he holds youth as the most sacred thing as the carrier of that very integrity, of our true nature.

Emerson suggests that nothing is more sacred than "the integrity of our own mind."

Emerson believed that people should not allow others to decide things for them.

Nothing is at last sacred but the integrity of our own mind. Absolve you to yourself, and you shall have the suffrage of the world. (p. 4)

If you focus on yourself, and care about yourself and what you think, then no one can control you.  If people believe in their own hearts, and do not let others think for them, then they will be successful.

Self-reliance was an important concept to Emerson because he believed that people were being told what to think.  To determine for yourself what you believe is the key to humanity.