What did the speaker mean by "He must have given the hand"?

One important element in Frost's "Out, Out—" involves agency. The poem opens with personified actions of the buzzsaw against the passive mountain setting in the distance. This juxtaposition of action and calm creates the context for other antithetical and seemingly paradoxical events.
When the buzzsaw cuts the boy's hand, the seemingly passive or disconnected speaker puts into language a horrifically active and invested action on the part of the mechanical tool. The speaker allows for ambiguity regarding what is responsible: the saw, the boy, or perhaps a cruel fate:

At the word, 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.


In a poem that contemplates the larger question of the meaning of this seemingly meaningless tragedy, these lines place emphasis on why terrible events happen and how one assigns blame or derives meaning. "He must have given the hand," the speaker insists hesitantly, for humans have a role in their destiny and in driving their tragic ends. How else could one bear the cruelty of a world in which a boy dies so suddenly? Of course, Frost complicates the speaker's claim by showing that even the speaker pulls back from this certainty, "however it was."

This is an interesting line. Here, the speaker may be attempting to determine whose fault the accident really was. He states that the saw, which is personified frequently in this poem, "seemed to leap" in the direction of the boy's hand. The speaker knows, however, and is seemingly reminding himself, that this is not really possible. The saw is not actually animate and does not have its own thoughts or feelings, which leads the speaker on to his alternative suggestion, that the boy "must have given the hand."
What he means is that, effectively, the boy must have offered his hand in the direction of the saw, whether consciously or otherwise. The detachment with which the speaker says "the" hand, rather than "his hand," adds a further level of distance between the boy and the blame—"the hand" becomes a separate thing, something almost sacrificed to the saw, rather than a part of the boy himself. The boy has held out his hand towards the saw, but in the moment of doing so, it ceases to be part of him as the saw claims it.