I like this poem. Although I haven't found any deep or symbolic meaning to it. I interpret it pretty literally - you know, a story about a boy who takes a special sword and kills a monster (kind of like this one series of video games that a few of us play....can't quite remember the name of it though...
What I like and admire most about this poem is Carroll way of mashing up words to make new
words. And I think its interesting that "chortle" is now "real word" used in our language today, and that "vorpal sword" found a home in role-playing games like Dungeons & Dragons.
But what this poem makes me think of the most is, back when I was in high-school, I was in the drama club. And the particular play we were doing called for 2 of my friends to wear costumes that were (how to describe it) "Sherlock Homes-ish" with those brown, plaid jackets, pipes, glasses, and those kind of golf caps. But each night after the show was over, immediately backstage, the two of them would recite this entire poem, taking turns every couplet, with British accents (or, at least their attempts at British accents). And they would intentionally over-dramatize thier lines like something out of Masterpiece Theatre. I thought it was really funny the first time I heard it, so it stuck in my mind. Fourteen years later, it's the first image I think of when I hear "'Twas brillig, and the slithy toves".