The Hebrew text is not meant to be taken literally. Only those who reject an education in college or who can't be bothered to research the past several thousand years of Judeo-Christian theology try to make it sound like it is supposed to be a literal historical account.
In fact, the first 11 chapters of Genesis contain the legendary history of Jewish nation. It isn't seen or made to be seen as literal history in any sense. That's why when you read it like history it sounds illogical. No one reads The Legend of Paul Revere as if its details are meant to be taken literal. And like that legend, it is not that the story has no historical basis (just as Paul Revere was a real person, and the legendary events in the poem took place, just not as detailed in the story), but the object is meant to be the lesson or moral of the story.
The story of the Tower of Babel is written in the format of a humorous anecdote. What it is really telling us is that human nature tends to be self-centered, that we as humans tend to think we can do great without any divine assistance, in fact without any god to guide and teach us. We can even get to the point of deifying our own reasoning and our own achievements.
In the story this is symbolized by all humanity working on a ziggurat (the Jews were poking fun at their heathen neighbors), believing they have made a tower so high that they would become famous. But God has to make an effort to "go down" and look at the folks and their tiny achievement. The story ends with God causing their languages to be changed so they cannot cooperate.
In fact, this probably is not explaining how different languages came about. This, like the story of Adam and Eve, is often misinterpreted by first blush and other elementary read-throughs of the text see in this type of storytelling as giving a definitive answer on "how" something "came to be." In fact, this type of mythology is about "what state we are in" or in other words explains a basic truth about society: humans cannot cooperate together for good because we are self-centered; we believe that our way is the best way. It also highlights that even if we all spoke the same language, humans can get so proud as to throw God aside and deify themselves and event their own "godless" government (the reason a King is involved in the story). Such attempts are always futile, the moral of the story says, because our self-centered behavior and thinking that our view is superior to others prevents us from "speaking the same language." This causes our hopes of unification, separate from God, to be useless and crumble, no matter how much we might seem to achieve at first.
Because people keep insisting that the Watchtower has it right to demand that the Scriptures be read the opposite way that the Jews and Christians intended them to be (and still do, according to most outside of the Watchtower) and like those blinded Jehovah's Witnesses agree to accept their interpretation over all others that the Bible has to be historical or false, the story looks unhinged.
Doing so just proves the point of the original intention of the story: if we try to do things the way God does not intend, substituting what we think is right, we end up with a whole lot of babel!
It's not the story that's unhinged. It's the mind of those who keep trying to squeeze non-JW literature into the mindset of the Watchtower's Governing Body.