Alliteration adds plenty of musicality and rhythm to the poem, emphasizing the speaker's nostalgia for the intense passion of young love.
For instance, multiple alliterative phrases in the first stanza help build a sense of anticipation as the speaker pulls out a shoebox filled with old letters. As the "lid lifts," the speaker catches sight of "their own recklessness written all over them." The lilting /l/ sounds work well to evoke the gentle removal of the box's top, while the gritty /r/ sounds seem to capture the roughness of the speaker's younger love.
In the next two lines, the quick patter of "pull their punchlines" and "fall flat" seems to at once suggest the elaborate prose of these letters and to evoke a kind of stutter, as if the "jokes" that once seemed so funny now feel awkward and ill-timed. And in the second stanza, /f/ and /tr/ alliteration brings a striking metaphor to life:
[...] Even now, the fist's bud flowers
into trembling, and the fingers trace each line and see
The insistent /f/ sounds here draw the reader's ear to certain details: to the speaker's curled "fist," a sign of pent-up aggression, and to the way it begins to "flower[]," to gently and beautifully unclench. The stuttering /tr/ of "trembling" and "trace" might then evoke the very "trembling" being described: the way the speaker moves haltingly, hesitantly as they revisit this old love.
Later, notice how the crisp /k/ sounds of "cardboard coffins" create a sharp, tight-lipped feel appropriate for a description of the letters being forever hidden away. Finally, the bold /b/ of the last line's "buried bones" is almost onomatopoeic, evoking both the "thudding" of the speaker's heart and the imagined thwack of a shovel against a grave.