I did a good chunk of research on this one, and all I got was anecdotal evidence. The tradition of animals being able to speak on Christmas Eve was new to me when my British wife mentioned it to me a few years ago, and indeed, most sources seem to agree it originates in the British isles. But the web is full of "my mother told me this," "I remember hearing it somewhere," and other noncommittal origins for the story. The Christian overlay is that it's meant to honor the animals who themselves honored Christ in the manger, but as always seems to be the case during these investigations of weird Christmas myths: yes, there are likely pagan origins.
Probably one of the more interesting interpretations comes from this Strange Horizons post from 2006, in which it's thought that the evenings before holy days/holidays (All Hallow's Eve, Christmas Eve, and, strangely unnoted by the Strange Horizons author, Walpurgisnacht) always seem to feature some form of thinness between worlds, or speaking with the dead, some other sort of prophecy, or the "one night in the year" that Supernatural Being X is allowed to do Supernatural Activity Y. Witches on Walpurgisnacht, the unquiet dead on Halloween... but surely innocent animals could not have the same level of uncannyness or even evil?
Let's be honest; sometimes the animals are not speaking of innocent things. Christmas in Ritual and Tradition, by Clement A. Miles, cites this story of a couple of ill-treated pets who speak and prophesy in Brittany:
Once upon a time there was a woman who starved her cat and dog. At midnight on Christmas Eve she heard the dog say to the cat, ‘It is quite time we lost our mistress; she is a regular miser. To-night burglars are coming to steal her money; and if she cries out they will break her head.’ ‘’Twill be a good deed,’ the cat replied. The woman in terror got up to go to a neighbour's house; as she went out the burglars opened the door, and when she shouted for help they broke her head.In the words of esteemed parapsychologist Dr. Peter Venkman... dogs and cats, living together, mass hysteria.