The idea of Romantic Escapism was really a reaction to the industrialized city life that became prevalent in American culture in the early to mid 1800s. The Romantics believed that in order for man to truly transcend everyday experiences to and live the true meaning and purpose of life. In order to put his ideas into action, Thoreau goes into the woods, literally, to Walden Pond, to live for two years and two months to experience a simple, streamlined life that will hopefully lead him to life's true meaning.
The first characteristic of Walden that is an example of Romantic escapism is that Thoreau literally leaves, escapes, the traditional day to day life he leads prior to his Walden experience. He travels into the woods to be solitary and to live life on his own terms.
Secondly, Thoreau uses this time to reconnect with nature, to remove himself from the city life and become more connected with the earth, himself, and, eventually a higher power. He lives off the land, builds his own home, grows his own garden and becomes deeply connected to natural events (like an ant battle and a bird taking off from a lake). He uses these seemingly innocuous events to find truth and meaning in his life.
Finally, Thoreau's experiences cause him to be even more self-reliant. He realizes that he wants to go into the woods so that he doesn't end his life realizing he never truly lived. He realizes through his solitary days that he wants to "suck the marrow" out of life and "live deliberately." The Romantics believed in the individual thoughts as superior to the masses, and Thoreau's time alone allows him to fully explore his own true self.