For the most part, change in language is gradual, though with some exceptions. People do develop new languages quickly, and not only with constructed languages like Esperanto or Klingon. It can happen naturally in a couple of ways.
When populations with different languages are brought into close contact, it's common for people to work out a "pidgin", a sort of half-language with vocabulary taken here and there from either parent language, but with minimal syntax and grammar. Or at least that's been a prominent theory. There's been some revisiting of just how simple pidgins really are.
One way or another, though, the children who grow up hearing the pidgin end up speaking a "creole", which is a full-fledged language with its own grammar and syntax, and distinct from either of the parent languages. Confusingly enough, these creoles are often referred to as pidgins. Tok Pisin is a classic example. It's been around long enough to develop its own dialects.
There are also sign languages that can be traced back to a small community that, as far as anyone can tell, invented its own language spontaneously since they needed to communicate and (obviously) couldn't use any of the spoken languages around. These are interesting in that they say something about our innate ability to use language, even without being exposed to one.
Unlike pidgins and creoles, which hybridize existing languages, these sign languages are constructed from scratch and, to be clear, they have the same kinds of complex features that spoken languages have, including the use of arbitrary symbols for abstract concepts. Like any other true sign language, these are not simple gestures and pantomime, and like other languages, sign or not, they too can develop variants and dialects over time.
There are probably other examples along similar lines. People have been using languages for quite some time, and there have been quite a lot of people over that time.
Nonetheless, the vast majority of the world's thousands of languages trace their lineage back over thousands of years of gradual change, likely all the way back to the exodus from Africa some 50,000 years ago, and beyond. Why should we think this?
The main reason to is that most of them bear a family resemblance to other languages. That is, they share features in common with those other languages, and those features indicate a branching pattern of languages diverging repeatedly from earlier forms (but also intermixing and hybridizing, so we don't have a pure "tree structure"). In some cases we can directly trace the development of a language family, for example the Romance languages of Europe (French, Italian, Portuguese, Romansch, Romanian, Spanish and several others, all derived from Vulgar Latin).
By careful comparison of the various features of similar languages, and by clues in the structures of the languages themselves, we can form a fairly precise model of how the various languages developed from a common ancestor over time, in a way that matches up well with written records where they're available, but does not require them.
If we know for a fact that some language families developed over time from a common ancestor, and most of the others show the same kind of resemblances to each other as members of well-documented families, the simplest explanation is that that what looks like a family resemblance is a family resemblance.
This is not to say that we can assert with certainty that, say, Bantu, Swiss-German, Aleut and Japanese share a common ancestor. Language changes quickly enough that the evidence becomes indistinct as we look at larger and larger families of languages. The most thorough and successful reconstruction of an ancestral language, Proto-Indo-European, takes us back only five or six thousand years.
To go back further takes careful statistical analysis, and this is not without its problems and controversies. However, there is no strong evidence that the languages listed above aren't related. They all seem to follow the same general plan, albeit with markedly different details. Again, the simplest assumption, until we know better, is that they all trace back to a small number of ancient languages, just as human ancestry traces back to a common ancestor (note that as always, "common ancestor" doesn't mean "first").