The Tower of Babel story is a fanciful attempt to account for a very real question: What was the first language and why are there now so many of them?
The video below from TED Ed shows a brief history of how languages evolve, as speakers of the same language lose contact with each other in the centuries after migration and gradually drift linguistically in different directions.
What's most interesting is not simply how we got multiple languages but rather how we determine, without the benefit of a time machine, which modern languages are related. To do this, historical linguists compare large numbers of words in different languages, looking for similarities that can't be explained by other factors, such as onomatopoeia (the word for cat is something like "miao" in several languages, but, well, there's likely an obvious reason for that) or borrowing (the word for tea in most languages is something like te or cha, but those can both be traced back to trade routes from different parts of China).
Similarities that are solid evidence of common ancestry may at first not look like similarities at all. For example, compare the English words father, foot, far, and five with the Ancient Greek words meaning the same thing: pater, podos, per (technically "forward"), and pente. Notice anything? The English terms all begin with an "f" sound while the Ancient Greek ones start with a "p" sound. When you piece together a whole series of systematic parallels like this across several languages (we can add in Latin pedes and German Fuß, both meaning "foot," for example), you can begin figuring out what the common ancestor, known as a proto-language, might have looked like.
The common ancestor of English, Latin, Greek, Russian, Gaelic, Hindi, and many other languages spoken in Europe and India is known as Proto-Indo-European, whereas the more recent common ancestor of just English, German, Dutch, Norwegian and the other Germanic languages is known as Proto-Germanic. The video below describes more of these systematic sound changes between Proto-Germanic and the rest of the Indo-European languages, and how they were discovered by linguists including the Brothers Grimm (yes, those Brothers Grimm). More in the video below.
We can do pretty well going step-by-step with this base-level comparison of languages—whether modern or those for which we only have written records—which has enabled linguists to devise around 50 proto-languages to varying levels of detail. But the real time-machine problem kicks in when we attempt to go even further back, back to what the common ancestor of these proto-languages might have been. Since there aren't any modern human societies that are incapable of language and any baby can learn any language, it's not unreasonable to suppose that we were probably using language when the first genetically modern humans began to spread throughout, and out of, Africa. But unlike cooking utensils or hunting weapons, languages don't leave physical artifacts of being spoken, and writing of any kind wasn't invented until somewhere between 50,000 and 300,000 years later. Ish.
And unfortunately, this means that any theory of the first human language must be based on pretty darn flimsy evidence. This problem was recognized as early as 1866, when the Linguistic Society of Paris prohibited further papers on the topic, and although this ban is no longer heeded, there's still nothing like consensus on where language came from or what the earliest ones might have sounded like.
But one tantalizing piece of evidence comes from a curious source: the newest languages of the world, like Nicaraguan Sign Language and other creoles, which arise when a group of children make order out of inconsistent linguistic input. We may never know for sure, but perhaps the process of creating a new language from scratch hasn't changed that much across the millennia.