According to the second paragraph in "Economy" in Walden, why has Thoreau decided to write about his life in Walden?
Thoreau was an adherent to Transcendentalism, an idealist philosophy which explored a spiritual connection between the inner consciousness and the external spirit that is nature. Ralph Waldo Emerson was also considered to be a Transcendentalist. He befriended and in some ways was a mentor to Thoreau. One of Emerson's most famous essays is "Self-Reliance." When Thoreau decided to live alone, some of these elements of transcendentalism and self-reliance were motivations.
Since he lived alone, if he were to write about that experience, he would necessarily have to write about himself. He writes:
Moreover, I, on my side, require of every writer, first or last, a simple and sincere account of his own life, and not merely what he has heard of other men's lives; some such account as he would send to his kindred from a distant land; for if he has lived sincerely, it must have been in a distant land to me.
In other words, Thoreau requires of every writer (including himself) to write sincerely. If a writer does write sincerely about his own life, it will be as if the reader is reading something new, something from a distant land. To live and write sincerely, as an individual, is to do things uniquely.
Thoreau also supposes that he writes for poor students. For any other readers, he hopes they will get what they can from his writing. Above all, Thoreau writes to explain his purpose for living as he did at Walden in the hopes to live up the expectation that he has set for all writers: to "give a simple and sincere account of his own life," something unique and individualistic, not concerned with what he has heard from other lives.
Nicely answered by Mr. Amarang9.