In poems, alliteration often connects words and ideas, or helps to create a musical sound. Because it's not very common in everyday speech, alliteration makes a poem's language sound elevated and distinctive. But it can also link meaningful words together, subtly strengthening a poem's meaning. Here it serves all these roles.
For an especially good example, take a look at the way alliteration works in the final stanza of "Rising Five":
The new buds push the old leaves from the bough.
We drop our youth behind us like a boy
[...] We never see the flower,
But only the fruit in the flower; never the fruit,
But only the rot in the fruit. We look for the marriage bed
In the baby’s cradle; we look for the grave in the bed;
This dense alliteration on /b/ and /f/ fits right in with the poem's final, insistent points. All those initial /b/ sounds are blunt and heavy, landing like blows on a drum, demanding that the reader pay attention—and they link ideas of new and old, connecting the fresh bud to the generative bough, and the young boy to the toffee wrappers he leaves behind. Something similar happens with the softer /f/ alliteration, which connects flower and fruit as one follows after another, over and over.
Part of what's going on here is to do with the poem's secret undercurrent of hope. While these last thoughts feel despairing, there's also a potentially uplifting double meaning: "rising dead" could mean "almost dead," but it could also mean "the resurrected dead"!
Mirroring that complexity, the alliteration here links ideas of before and after, cause and effect, and past and future, suggesting that even though humans have a hard time staying in the present moment, there's always a quiet hope of renewal. Even dead ends might be connected to new life.