If you've taken a peek at our "Form and Meter" section, you'll remember that we mentioned in there (somewhere) that this poem really sounds like an internal monologue of sorts—kind of like something you would say to yourself in your head, or even out loud while alone in your room. Part of the reason for this is because the poem is all about silence, and about how the speaker grieves in silence. In other words, it's like the speaker says "Hey, I'm sad, but I'm sitting here in a corner and not telling anybody about it."
So, basically what we're saying is this poem sounds like a poem that plays in the speaker's head while others around him are talking about his old flame. It often unfolds, sonically that is, as a series of quick, semi-connected thoughts not fully articulated in the way they would be, say, in an essay (with complete sentences and all that). Take the poem's first four lines:
When we two parted
In silence and years,
Half-broken-hearted
To sever for years. (1-4)
Say them out loud to yourself. Say them again. The bang-bang-bang of lines 2-4 in particular sounds like a swift succession of ideas, doesn't it?
Now, although we've established the "internal monologue, quick succession of thoughts" aspect of the poem, we need to, or can, refine it just a tad. This poem obviously rhymes, and does have some structure, albeit a loose one. In some ways, this poem is like slam poetry done in private. What's slam poetry? Well, it's sort of impromptu rapping, or poeticizing (and it usually takes place in some artsy café). Read about it all right here. Sometimes, this poem sounds like a poem made up on the fly, a series of poetic ideas or thoughts very loosely knit together through its sounds, in addition to its ideas.
The most obvious way in which this process can be seen at work is in the poem's frequent repetitions of sounds. Sure we have the rhymes, but this poetic rapper is a little more clever than that. Take lines 17-20:
They name thee before me,
A knell to mine ear
Shudder comes o'er me—
Why were thou so dear?
Note, for example, the assonance of the long E ("thee," "me," "ear," "me," and "dear") and the long I sounds ("mine" and "why"), as well as the consonance of the long R ("before," "ear," "shudder," "o'er," "wert," and "dear"). These kinds of chiming sounds are used all the time through the poem, in fact. If you were trying to "rap" on the spot, or make up a poem really quickly, you would certainly try to be clever and repeat sounds, wouldn't you?