An abundance of alliteration lends the poem rhythm and musicality, making it feel more intensely dramatic. Take a look at the first four lines, for instance, where /l/ and /be/ alliteration start the poem off with a flourish:
Lo! 'tis a gala night
Within the lonesome latter years!
An angel throng, bewinged, bedight
In veils, and drowned in tears,
These intense sounds are fitting for a poem that presents all of human history as a gruesome tragedy put on for angels at some heavenly "gala"!
In lines 9-12, /m/ alliteration evokes the muffled sounds of unspeaking actors moving back and forth across the stage:
Mimes, in the form of God on high,
Mutter and mumble low,
Those muted sounds (supported by the /m/ consonance in "mimes" and "mumble") evoke the muffled quiet onstage, helping the reader share the suspense the angels must be feeling while watching this "play."
Whispery /s/ alliteration and more general sibilance in lines 19-22, meanwhile, emphasize humanity's utter lack of progress. The speaker says that ultimately humans always end up returning "To the self-same spot." Here, the hiss of the /s/ sounds evokes the speaker's scorn for human beings, who never seem to learn from their mistakes.
These are only a few examples of the copious alliteration throughout the poem. Everywhere alliteration appears, it contributes to the speaker's melodramatic, hyperbolic tone, making this tale of universal woe and despair feel that much more theatrical.