The bulk of the story is told from Hank's point of view, and as the main character, he's right in the middle of all the action. This also means that we see things filtered through his perceptions, which helps Twain make fun of the trappings of Arthurian legend. Hank can say how stupid a knight looks in his armor, for instance, which helps Twain both make his point while also giving us an idea about the kind of guy Hank is.
A few times at the beginning and end we shift to a different kind of first-person narrator. The unknown Warwick Castle visitor opens and closes the books, and Clarence steps in briefly to tell us how he hides Hank's slumbering body. In both cases, these narrators deliver much-needed information to us that we can't get from Hank's perceptions alone.
The text places a premium on hardheaded practicality after all, and at certain points, it becomes impossible for Hank to write himself. He can't actually write at the end when Merlin lays him low—"The Boss has never stirred—sleeps like a stone" (44.3)—and then back in the present when he lies dying: "He lay muttering incoherently some little time; then for a time he lay silent, and apparently sinking away toward death" (45.2). So Clarence and the narrator do it for him, and Twain uses the framing device to make these narrative shifts make sense.
It's a little awkward shifting back and forth in the beginning, but otherwise, Twain would be guilty of the same wonky logic of the Arthurian stories he's trying to skewer… and he can't have that.