Alliteration, like its cousins assonance and consonance, gives this poem both music and meaning. Besides simply sounding good, repeated sounds evoke this poem's world and draw attention to important moments.
For instance, take a look at the sibilant alliteration in lines 3-4:
And pardon that thy secrets should be sung
Even into thine own soft-conched ear:
The alliterated /s/ sounds just like the whispered secrets it describes—and like the softness of Psyche's ear, a vividly intimate image that suggests how close the speaker is getting to the goddess he praises. It's as if his lips are brushing her skin as he speaks.
Later on, alliterative /m/ sounds follow the speaker as he goes deep into an imaginative trance:
So let me be thy choir, and make a moan
Upon the midnight hours;
That repeating /m/ makes the speaker's words sound like a chant—and links that "moan," the sacred song, to the "midnight," so that the song and the night both seem enchanted.
Since it sounds a little different than everyday speech, alliteration can also direct the reader to meaningful moments. That "pale-mouth'd prophet" who turns up at the end of the third and fourth stanzas is already an unexpected, striking image—and the repeated /p/ sound draws even more attention to him.