What does Ralph Waldo Emerson mean when he says "And we are now men, and must accept in the highest mind the same transcendent destiny?"

To understand this line from "Self-Reliance," we have to put it in context. Emerson is urging the reader to listen to his own intuitive insights, and not put them aside because he doubts his ability to discern the truth.

"A man should learn to detect and watch that gleam of light which flashes across his mind from within, more than the lustre of the firmament of bards and sages. Yet he dismisses without notice his thought, because it is his."

He points out that both children and "great men" have in common a confidence in their own intuitions and insights:

"Great men have always done so, and confided themselves childlike to the genius of their age, betraying their perception that the absolutely trustworthy was seated at their heart, working through their hands, predominating in all their being."

This leads to the line in question. Emerson has explained what great men do. They trust in themselves, like children. Once, we were children, and so we possessed this trait of confidence and self-trust.

"And we are now men, and must accept in the highest mind the same transcendent destiny…"

In other words, as adults, we must embrace this childlike trait, and employ our most elevated form of thought to achieve transcendence -- going beyond the limits of the concrete world to appreciate the higher spiritual reality.
To underscore this message of reconnecting with one's childlike self-reliance, Emerson talks further about this virtue in boys. They aren't as oppressed by the pressure to censor themselves. They don't restrain themselves from saying or following through on what they believe, and this, says Emerson is a "healthy attitude of human nature." It's only later, as we grow up, that we stop listening to our inner voices.

"These are the voices which we hear in solitude, but they grow faint and inaudible as we enter into the world. "

And that, says, Emerson, is antithetical to our true purpose. God meant us to trust in our truthful, individualistic voices, and follow them to discover what really matters.