This book is essentially different from any other that has been published concerning the "late war" or any of its incidents. Those who have had any such experience as the author will see its truthfulness at once, and to all other readers it is commended as a statement of actual things by one who experienced them to the fullest. – John Ransom's Andersonville Diary
What is up with this epigraph? For the answer to this one, let's jump back to another major war in American history—the Civil War. During that war, many Union soldiers (the guys from the North) kept diaries while they were in Andersonville, a Confederate prisoner of war camp, and those accounts were aptly called the Andersonville Dairies.
The one kept by John Ransom included this anonymous note—the epigraph we're referencing—at the beginning.
As an epigraph to The Things They Carried, it works with the dedication (discussed in "Quotes: Truth") to reassure us right off the bat that the stories we're about to read are absolutely true. Later, of course, we realize that the stories are true for O'Brien's definition of truth... which we may have difficulty understanding at first.
Once we've reached that point in our reading, the epigraph reassures us once again that while we, civilians, may not understand how the events of the book can be both fabricated and true, other soldiers will get it, and we should take their word for it.
It's a little defensive, true, but it also gets us thinking right away about truth, and eventually about the communication gap between civilians and soldiers—ideas that are absolutely essential to understanding the book.