The poem begins with nonsense words. Nearly every noun, adjective, and verb is an invention of Carroll's ("gyre" being the exception), so our initial engagement with the poem happens through its appealing sound rather than its more specific meaning.
Yet even in these initial lines, that sound evokes the sense of an adventurous epic through strong, full rhymes and regular iambic meter. Indeed, this stanza establishes the rhyme scheme that the rest of the poem will follow (for the most part)—an ABAB pattern—as well as the poem's iambic tetrameter. Despite not knowing what the words actually mean, they fall easily into the familiar da-DUM rhythm of an iambic poem:
'Twas brillig, and the slithy toves
This exact rhyme scheme and meter mimic the style of a lyrical ballad, a poetic form that most often tells a story of love or adventure (in this case, the latter). Carroll thus makes it clear to the reader that this is the beginning of some epic story—quite a feat, considering that so few of these words are foreign to us.
The rest of the stanza is filled with more sonic lushness that makes the words' specific meaning rather, well, meaningless. The word "brillig" shares assonance with the /i/ sounds of "mimsy," while the words "gyre," "gimble," "borogoves" and "outgrabe" each have alliteration and consonance with /g/ and /b/ sounds. "Borogoves" and "mome" are yet another pair of assonance. All these repeated sounds make the poem seem carefully crafted; it comes across as highly literary and poetic, befitting of its ballad form. Of course, given that the poem is also nonsensical, this could be thought of as a good-natured satire of that form: all pomp, no circumstance.
Indeed, since it is difficult to tell what's actually happening in this stanza without the contextual aid of Through the Looking Glass, the poem begins in almost utter mystery. We have trouble knowing what to expect (in fact, we might say the nonsensical quality of the poem defies expectation), but the familiar form of the ballad and the whimsical, creative sounds keep us engaged and curious.