On the most basic level, alliteration adds sonic interest and music to the poem (we've highlighted some specific examples of alliteration here). It often mirrors the literal music that the speaker hears, in fact, and in doing so makes the poem all the more immediate and vivid for readers. Take, for example, the “cracked cornets” from line 31, the sharp alliteration of which reinforces the idea that a song “hammers” in the speaker’s head.
As with other forms of repetition, the reappearance of many similar sounds also makes the speaker and the lady’s interactions seem all the more predictable. Hearing the same sounds over and over again evokes the repetitive nature of their lives. In fact, alliteration actually appears within the two lines that the poem repeats directly, which describe each character’s typical routines:
Let us take the air, in a tobacco trance
I shall sit here, serving tea to friends
Alliteration also just makes phrases like this more memorable, making them linger in the reader's mind. Consonance enhances the effect; note how sharp /k/ sounds echo in "take"/"tobacco," and how the /s/ sound of "trance" continues into the next line with "sit"/"serving."
Finally, alliteration can place additional emphasis on certain words and phrases. Take the lady's dialogue in line 19:
"You do not know, you do not know / How much they mean to me, my friends,
The clear alliteration here evokes the lady's sheer insistence on her point (that is, that friendship is deeply important to her).