The letter, printed (as others have noted) in The New York Daily Tribune on 22 August 1865 ("Letter from a Freedman to his Old Master," p. 7), is prefaced with, "The following is a genuine document. It was dictated by the old servant, and contains his ideas and forms of expression. -- Cincinnati Commercial."
I've no particular reason to doubt that this letter is what it's claimed to be -- a letter written by a former slave to his former owner -- but it bears noting that a third party had a hand in its creation, at least to the extent that he or she took down Anderson's words. So, perhaps the writing also reflects corrections and revisions made by the person to whom Anderson dictated the text, though Childs prefaces her reproduction with "[w]ritten just as he dictated it." (How she knows this is unclear.)
Also unclear is how this letter arrived at the Cincinnati Commercial; given the time between the penning of the letter (August 7) and its appearance in the Cincinnati paper (sometime before August 22), a copy of the original (if the original had been sent) must've been handed over by Anderson, the person to whom he dictated the letter, or some advocate on either's behalf, and not forwarded by someone in Big Spring, Tennessee. (I should note that it would be interesting to see the letter that elicited Anderson's response.)
Text Source: snopes.com
Image Source: Mississippi Beautiful