Alliteration helps brings the poem's images to life at several powerful moments.
The first example, in line 1, is all about the element of surprise. It's dawn, and the speaker observes a moment of uneasy calm on the battlefield as the "darkness crumbles away." Suddenly, a rat appears and interrupts the speaker's train of thought: "a live thing leaps my hand" (line 3). This alliteration is lively and surprising, like the rat itself.
In the same sentence, the speaker "pull[s] the parapet's poppy" to tuck behind his ear, apparently as a good-luck charm. This alliteration (and consonance) is cheerful, almost childlike, as though borrowed from a nursery rhyme. Yet it also gently suggests violence through all those (ex)plosive sounds.
Alliteration appears again in line 18: "fields of France." Here, the /f/ sounds are soft and pleasant. Taken in isolation, the "fields of France" sound like an attractive place to be! But the ugly adjective "torn," with its hard /t/ sound, contrasts with the /f/s and suggests the violence occurring in those fields.
The speaker then asks the rat (via apostrophe) what it sees in the eyes of the soldiers as artillery and gunfire are "hurled through [the] heavens." The alliterative, breathy /h/ sounds suggest physical effort—the kind of huffing and puffing you make when you hurl something.