Many of Emerson's and Thoreau's beliefs were formations and initiations of Transcendentalist philosophy. The philosophy espoused elements of Kant's idealism and the Romantic's love of nature. Transcendentalists directed their philosophical inquiry inward, focusing on the self more than on social institutions and the empirical observations of the material events in the world. But they also focused on natural things in the world as as part of one differentiated, living system. So, they were idealists (thinking abstractly) but they were naturalists (balancing that introspection with a scientific and spiritual connection with nature.) Thus, they transcended the barrier between self and nature. God, nature, meaning, and each self are all parts of the same system accessed by this transcending barrier (therefore, a bridge). Accessing meaning and truth in this way meant that one does not have to go through an authority or a social institution, like one would have to go through a priest to commune with God.
Consequently, you could also say that Emerson's and Thoreau's beliefs in social reform are effects of their transcendental principles, especially how they favor individualism over social institutions.
In "Nature," Emerson writes:
The lover of nature is he whose inward and outward senses are still truly adjusted to each other; who has retained the spirit of infancy even into the era of manhood. His intercourse with heaven and earth becomes a part of his daily food.
A deep connection with the self and that 'self's' connection with nature, on a more fundamental level than merely seeing and hearing nature, are hallmarks of transcendentalism. "Finding truth in yourself" is a cliche but apt description of transcendentalist thought.
In "Self-Reliance," Emerson writes:
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.
In "Solitude," fromWalden, Thoreau writes:
I go and come with a strange liberty in Nature, a part of herself. As I walk along the stony shore of the pond in my shirt-sleeves, though it is cool as well as cloudy and windy, and I see nothing special to attract me, all the elements are unusually congenial to me.
In his solitude, Thoreau feels more welcomed (by Nature) and feels he fits in more naturally than he would in the city. Therefore, he can be alone in nature but not feel loneliness. The transcendentalist goal is to be the bridge (/) between self and nature.
Broadly speaking, Transcendentalism was characterized by individuality, a strong belief in the human spirit, and the idea that people ought to obey the dictates of their own conscience. Emerson argued that philosophy, art, and morality ought to be derived from one's own soul, which was connected, in a sense, to that of all mankind. Thoreau also preached a morality derived from one's own conscience, which was ultimately what men were bound to obey. This is most evident in his essay "Civil Disobedience." Additionally, Thoreau's self-imposed mission of simplifying his life by living an ascetic existence at Walden expressed a concern common to Transcendentalists that modern life was corrupting the human spirit.