Robert Frost was a master of matching sound to sense in his poems. It's no surprise, then, that this poem about "sound" (a word it mentions twice!) contains alliteration that reinforces its meaning.
In particular, the poem is full of /s/ and /w/ words throughout. Look at how many crowd the first six lines alone:
There was never a sound beside the wood but one,
And that was my long scythe whispering to the ground.
What was it it whispered? I knew not well myself;
Perhaps it was something about the heat of the sun,
Something, perhaps, about the lack of sound—
And that was why it whispered and did not speak.
These soft, sibilant /s/ sounds and liquid /w/ sounds mimic the "whispering" of the scythe itself. (Notice that the words "whispering" and "whispered" also contain /s/ consonance in the middle; they could be considered onomatopoeia words.) In effect, the whole poem takes on the sound of the scythe.
Other alliterative phrases crop up, too, as in lines 10-11:
To the earnest love that laid the swale in rows,
Not without feeble-pointed spikes of flowers [...]
Here, the repetitive sounds might evoke the scythe's repetitive movement as the speaker mows or "la[ys]" the grass down, as well as the recurring sight of "flowers" along the way. However, there's not always a neat and clear match between alliteration and meaning. More generally, the device gives the poem a pleasing "sweet[ness]" (see line 13)—a hushed, lyrical quality that fits its celebration of the speaker's quiet work.