Part of the answer to this question lies in how we define evolution. At its most basic level, evolution merely means “change.” If this is the meaning of evolution we adopt, that it is quite clear that human languages have evolved. After all, anyone who has read Shakespeare would acknowledge that the English language has “evolved” over the past four hundred years! The problem is that by defining evolution in this sense we are not making a particularly significant claim about human language.
What usually seems to be meant when the evolution of language is discussed is the idea that language has evolved in a Darwinian sense. That is to say, human language originally was quite simple but became increasingly complex over time. In this view, human language at first was not much more than the sort of vocalizations used by some primates, but somehow became something more than that.
Superficially, this view of the evolution of language would seem to have a lot of support, particularly if we think of linguistic evolution as being analogous to biological evolution. For one thing, it is obvious that like new languages develop from older languages, like a new species of animal is thought to evolve from an earlier existing species. For instance, the so-called Romance languages of French, Italian, Portuguese, Romanian, and Spanish developed from Latin, which can be seen from the similarity of the French word for “window”--fenetre—to its Latin equivalent--fenestra.
Moreover, like biological species, languages can become extinct—think of languages like Hittite or Etruscan. Some 19th century linguists went so far as to posit that there is a sort of “survival of the fittest” among languages, the “fittest” languages, naturally, being the European languages spoken by these scholars! It is tempting to think this is true—one might argue that the reason English has become the most widely-spoken second language in the world is its large vocabulary (the largest of any language I believe) and its flexibility in developing or adopting new words. However, though there might be some truth to such a notion, it would seem that a more obvious explanation of the dominance of English is the global political and economic ascendancy of two English-speaking countries—Great Britain in the 19th century and the United States in the 20th century.
Furthermore, languages definitely do not evolve in one sense assumed in the Darwinian paradigm: changes in languages do not seem to have any particular direction, unlike biological species which are thought to evolve in ways that serve the purpose of the species’ survival. For example, English used to be an inflected language, like Latin or modern German. That is to say, in English it used to be the case that changes in the grammatical functions of words were signaled by changes in their form. Today, however, English is primarily a word order dependent language, meaning that the grammatical functions of words are determined by their position in a sentence. Thus, “Mary” has two different grammatical functions in the sentences Mary kissed John and John kissed Mary depending on its position in the sentence. However, this change in the nature of English syntax (the structure of language) does not seem to be motivated by anything. One might argue that it has been English easier to learn, but that is looking at the matter from the perspective of someone who is a speaker of a word order dependent language. Someone who speaks an inflected language might see things differently.
Perhaps most damaging to the idea that human languages have evolved in a Darwinian sense is the fact that there is no evidence for the earlier existence of languages that were much simpler than modern languages. In the Darwinian paradigm, the initially simple becomes complex. However, my understanding is that even the oldest languages of which we have written records, like Sanskrit, are no less complex than modern languages like German or Chinese. Even the languages of so-called primitive peoples in the modern world are hardly that simple in character. Linguists have been able to reconstruct the ancient Indo-European language, the ancestor of modern European languages, despite the lack of written sources for it; nonetheless, it was apparently not that simple in nature either.
Given this lack of evidence for the evolution of human languages, evolutionary linguists are left to argue that at some point early in human history, language must have been simpler than it is now, but this can only be true if the Darwinian view of evolution is valid. Otherwise, this is purely speculation, no matter how impressive some of the speculation may seem.
In short, we cannot really speak of the evolution of language as a fact, unless we mean it in the broadest sense—that language has changed over time. However, this is a rather trivial claim and besides, it does not seem to be what evolutionary linguists mean when they speak of the evolution of language. Consequently, I would argue that we must treat any claims about the evolution of evolution with some skepticism.
Image: Breughel's painting of the Tower of Babel