In the very beginning, Emerson personifies Nature:
Nature says, -- he is my creature, and maugre all his impertinent griefs, he shall be glad with me.
This connects humanity with nature as if we exist as equals, as if we were dependent on one another.
He uses rhetorical questions to influence the reader.
Why should not we also enjoy an original relation to the universe? Why should not we have a poetry and philosophy of insight and not of tradition, and a religion by revelation to us, and not the history of theirs?
These questions challenge us and encourage us to liken ourselves to generations of the past.
He uses oxymoron or great paradox:
I am glad to the brink of fear.
Fear doesn't make us glad or vice versa. But when in Nature, I believe he is expressing that the ultimate benefit of being there is experiencing emotion at it's greatest extremes.
He uses simile and metaphor:
In the woods too, a man casts off his years, as the snake his slough, and at what period soever of life, is always a child. In the woods, is perpetual youth.
Comparisons like these are important because when readers can relate to things, they have a greater ability to grasp what a writer is saying.
Much of the above is in reference to your question about literary elements. In terms of style, he fits the style of an essay, but he uses a first person point of view which doesn't always happen in essays.