The alliteration in "Eat Me" helps to evoke the speaker's appearance, her partner's obsession with her weight, and the difference between their two experiences of this abusive relationship.
The first examples of alliteration appear in the third stanza:
Then he asked me to get up and walk
round the bed so he could watch my broad
belly wobble, hips judder like a juggernaut.
The round /b/ sounds here evoke the speaker's own roundness (and chime with the /b/ in "wobble"); the letter even looks like a belly. "Judder" and "juggernaut"—which are also assonant—conjure a picture of a jelly-like wobbling, which is exactly what the speaker's partner wants to see.
Soon after this stanza, the partner talks about his desire for fat women. "The bigger the better," he says in line 10, the alliteration making it seem like a pithy saying that he uses all the time (and chiming with the /b/ in the previous stanza). And the repeated /m/ sounds in "Multiple chins, masses of cellulite" in line 12 feel dense, evoking the "masses" of flesh the partner fetishizes.
Alliteration also draws attention to some meaningful comparisons between the speaker's and her partner's different experiences. For instance, the speaker's "only pleasure," for many years, is "fast food," while her partner likes to see her "swell like forbidden fruit." These two moments of /f/ alliteration invite the reader to notice that, while the speaker can only take pleasure in cheap "fast food," her partner relishes the "forbidden fruit" of her body as if he were in Eden itself. Her life thus feels sad and limited compared to his fantasy-world.