First thing to do is understand romanticism and transcendentalism.
Transcendentalism was an idea that basically says you need to do what your own conscience tells you is right. Never mind what society tells you, just do what you believe in.
In this context, the main point of Romanticism is that emotion is superior to logic and intellect.
With that in mind, you should be able to see these things in the movie.
Keating encourages the boys to see things from their own perspective. He encourages them to climb up on their desks, to go outside the classroom, to look at things differently. He encourages that one boy to act in the play, no matter what his dad says.
When the boys rip the boring essay out of the book, they are approaching poetry from an emotional point of view rather than from the essay's more logical point of view. When the one boy writes the love poem to the girl he is going on emotion.
The movie shows people being encouraged to do their own thing and to have feelings rather than just logic. These things are right in line with Transcendentalism and Romanticism.
There are some indications that tip off the viewer that Romanticism and Transcendentalism are highly evident throughout. The facts that "Keating" is a form of "Keats," the Romantic poet, is something that cannot be escaped. His continual mention of "seize the day" or "Carpe Diem" is a Romantic idea that Keats, as well as the other Romantic thinkers, emphasized as a major component of their work. "Gather ye rosebuds while ye may" is a Romantic tenet in that it stresses that individuals fully immerse themselves within the natural setting in the expression of self in a singular moment. These themes are taught, along with the reverence of past poets, nostalgia for the society, in general, and emphasized throughout the film. Transcendentalism closely follows when Mr. Keating asks the students to call him "O Captain, My Captain" from Whitman's poem. The invocation of America's poet, a Transcendentalist who sought to link nation, emotion, and art together is of critical importance. Keating's mission as a teacher seems to be to fuse these elements in the boys' education. In tearing up the Pritchard volume of poetry, this is a highly Transcendentalist notion in its assertion of individuality and distinctive voice, breaking from text and traditionalism. This reminiscent of the self- reliance that was articulated through the works of Thoreau and Emerson, and the fierce rejection of "Traditionalist" study that is seen in the poems of Whitman.