Alliteration appears throughout "At an Inn," filling the speaker's lines with music and emotion. The speakers feelings are quite intense—as though they've been bubbling away in a pressure cooker—and the poem's frequent alliteration evokes this intensity. Take lines 1-4:
When we as strangers sought
Their catering care,
Veiled smiles bespoke their thought
Of what we were.
The alliteration starts the poem off on a bold, forceful note in terms of sound, already suggesting the speaker's longing for "what we were" (that is, for closeness with this friend). The hard /c/ "catering care" specifically hints at the eager enthusiasm of the staff at the inn, as does the later /m/ sounds of "Made them our ministers / Moved them" in lines 13-14.
"Swift sympathy" has a quick, slippery sound, while "living love" is warm and luxurious. The inn staff are keen to help the newcomers because of the love they falsely perceive, and these examples of alliteration ring hollow; they grant the poem a gentle, pleasant beauty that evokes the glow of love, but because love isn't actually there, the sonic effects are a kind of fantasy.
By contrast, note the plosive alliteration in lines 21 to 24, with its bold /b/ and /p/ sounds:
But that which chilled the breath
And palsied unto death
The pane-fly's tune.
Notice how these sounds require the reader to push air out of the mouth, creating a mini version of that "chill" that blows between the speaker and his friend. It's a totally opposite effect from the pleasant /l/ sounds of "left," "Love," and "love-light," marking the shift from fantasy to reality.
The poem uses alliteration in the end to build to its rhetorical height. The poem ends on a strong, high note of woe-is-me, and alliteration works alongside apostrophe and repetition to make this a dramatic, heartrending moment. The sibilance of "severing sea" also subtly evokes the salty splash of the waves that keep the speaker and his now-beloved apart.
Finally, note that much of the poem's alliteration comes in pairs, almost mocking the human pair that never was (the speaker and his friend). "Strangers sought," "catering care," "swift sympathy," "living love," "love-light," "come [...] Came," "his hold," "love lingered," "severing sea," "land [...] law," :"stand [...] stood"—these are all alliterative couplings! It's like the speaker sees couples wherever he goes, and speaks them whenever he opens his mouth. Love—or the lack of it—is on his mind, expressing itself in everything he does.