Tennyson's poem, overall, sounds quite musical and lulling (with some exceptions—more on that below), and takes advantage of multiple sound effects to get us there.
Assonance
He uses assonance both to provide musicality to his poem and to emphasize or create meaning. Here's one example from lines 257-260:
Calm on the seas, and silver sleep,
And waves that sway themselves in rest,
And dead calm in that noble breast
Which heaves but with the heaving deep.
Notice how Tennyson repeats the long and short E sounds in the first line, and then the long A sound in the second, followed again in the last two lines with the long and short E sounds. The repetition of these sounds replicates the lulling motion of the sea, which emphasizes the idea of calmness in the stanza.
Let's look at another example from lines 709-712:
The moanings of the homeless sea,
The sound of streams that swift or slow
Draw down Æonian hills, and sow
The dust of continents to be
Check out the repetition of the long, short, and rounded O sounds throughout this stanza. Try reading it aloud and really drawing out those vowels. Do you start to hear yourself moaning? Like "Ooooooh...and Oooowwwwww...and Aaaaahhhhh"? That's basically the effect. And it makes sense, right? He's talking about the sound the sea makes, and it's a mournful sound (it's "homeless," after all, which is pretty sad).
Consonance
Not wanting consonants to feel left out, Tennyson also uses plenty of consonance, which also contributes to the musical sound and the poem's meaning. Here's when Sorrow addresses Tennyson: "'The stars,' she whispers, 'blindly run; / A web is woven across the sky" (81-82). The use of the repeated soft S sound recreates the soft hiss of whispering, providing a lulling sound, but also suggesting closeness or confidentiality between these two figures.
Alliteration
Where consonance is repeated consonant sounds in various places in a word, alliteration repeats initial consonant sounds.
In the stanza where Tennyson talks about the "Satyr-shape" has "bruised the herb" and "bask'd and batten'd in the woods" (722-724), the repetition of the harsh B sound provides a hard and rough rhythm that replicates the violent act of bruising and battering things.
Enjambment
While most of Tennyson's lines are heavily end-stopped, several use enjambment. This breaks up the lulling music of the poem and signifies where Tennyson's thoughts are getting out of his control—where they're escaping the bounds of the poetic lines. He's dealing with some pretty heavy-duty stuff, so it's not surprising that his thoughts get away from him.
One place we see this is when Tennyson is struggling with his ideas on religion and science (or reason) in lines 2437-2440:
Who loves not Knowledge? Who shall rail
Against her beauty? May she mix
With men and prosper! Who shall fix
Her pillars? Let her work prevail.
Tennyson doesn't stop at the end of these lines; the end-stops come right smack in the middle, which breaks up the usual flow of the poem. It makes sense that this breaking of poetic bounds would occur when he's wrestling with something so compelling.