The situation surrounding "who's talking?" in this poem is a bit of a mystery. In Through the Looking-Glass (which happens to be the actual book where "Jabberwocky" was published), Alice simply finds the poem in a book on the Queen's table. So the poem has already been published in the story in which it's published.
What this means for our estimation of the speaker is that the context of the poem, i.e., being part of Through the Looking-Glass, isn't going to help us much, because Alice can't make heads or tails of it either. So now what?
Well let's see. Think of our speaker as a guy telling a story around a campfire. A story that he's heard from someone, who has heard it from someone else, who heard it from some other guy. This fits right in with the poem being in a mysterious Anglo-Saxon and bardic tradition. It's all about the oral history.
The monkey wrench here is that our speaker isn't from any land that we know. He knows about animals called borogoves and bandersnatches. A good way to think through what our speaker might be like is to think about, say, the Hobbits and Dwarves in Tolkien's Lord of the Rings. These characters are mostly human, and they use speech that we can recognize as English, but they come from another realm altogether.
So our speaker comes from a place that's like England (which is where Carroll was from), but at the same time not really at all. Much like the Shire from Lord of the Rings.
Stylistically, this guy is a world-class storyteller. Remember those sleepovers you used to have when you were little, when you'd all take turns shining a flashlight up your nose and you'd try to tell the spoooookiest story you could? Minus the ghosts, that's what our narrator is trying to do here – "and then…OH NO A MONSTER!" And we all jump back in our seats before leaning forward, wanting desperately to see who wins out in the end. Our narrator takes great pleasure in the suspense of storytelling, building us up and telling us with great relish this story of mighty victory, before setting us gently down where we started.