Frost uses a detached and emotionless tone to create a sense of horror in this poem. The closest that the narrator gets to expressing emotion is when he comments that he wishes "they might have said" to "Call it a day" and given the boy a half an hour off (in order that he might have avoided the accident completely). His description of the, frankly, horrifying accident is so devoid of extremes that we can almost misunderstand what happens. It is presented, somewhat ironically, in a completely unhorrifying manner. The narrator says,
At the word [supper], the saw,
As if to prove saws knew what supper meant,
Leaped out at the boy's hand, or seemed to leap--
He must have given the hand. However it was,
Neither refused the meeting.
He describes the saw cutting off the boy's hand (or nearly so) as a kind of meeting between well-intentioned, polite strangers, rather than as the completely blood-soaked horror show that it must have been. He employs a kind of understatement with this description, personifying both the saw and the hand as willing acquaintances.
The description of the boy's attempt to "keep / The life from spilling" is another strangely calm and almost poetic way to describe what happened. It employs metonymy as well, associating the blood that spills from his wound with the boy's life, as we certainly need our blood to live. Presumably his extreme loss of blood is why he dies in the end. This metonymy removes anything graphic from the description and makes it sound almost lyrical or beautiful.
In the end, then, the juxtaposition of the author's detached tone and the narrator's calm descriptions with the bloody and violent reality create a kind of horror. Why isn't the author or the narrator, or even the other people in the poem more upset?! The reason gets at the poem's theme: death is commonplace -- even when it happens so tragically and suddenly to one so young -- and life must go on. There is, honestly, something horrible about this idea too: that the extinguishing of life is so common that it ceases to produce any real effect.