The repeated initial sounds of alliteration—among other kinds of repetition—play a big part in this poem's melody and meaning. The frequent alliteration here helps to draw connections between important words, and to create the poem's rolling, musical sound.
Look no further than the famous first lines for a clear example:
We are the music makers,
And we are the dreamers of dreams,
Here, alliteration links up "music" with the "makers" who create it, and "dreamers" with their "dreams." (The connection between "dreamers and "dreams" is also an example of polyptoton—more on that under "repetition.") Those linkages begin a pattern that will continue all through the poem, in which artists and their art seem so closely connected that it's sometimes hard to see a separation between them.
In some spots, alliteration makes images feel all the more forceful and urgent as well. Take lines 14 and 16, where the hard /c/ and /k/ sounds imbue the line itself with power and strength:
Shall go forth and conquer a crown;
[...]
Can trample a kingdom down.
The effect here is even clearer when considering that the same sound repeats within the word "conquer" itself (an example of consonance).
Repetitive sounds also fit in with the poem's ideas of artists as "music makers" more generally: they make these speakers' voices sound musical. Take a look at the alliteration in lines 65-68:
Great hail! we cry to the comers
From the dazzling unknown shore;
Bring us hither your sun and your summers;
The melodic, attention-grabbing repetition of initial /c/ and /s/ sounds here helps to bring the poem to its climax.