As is often the case with idioms, no one's positive of the origins of "cat got your tongue." However, there's no shortage of theories. Here are a few, in no particular order...
Theory #1
The saying comes from the Middle East, where as punishment, liars had their tongues ripped out and fed to the king's cats.
Theory #2
Fear of a whipping with a cat-o'-nine-tails, or "cat" for short, could paralyze a victim into silence.
Theory #3
The expression comes from the Middle Ages when witches were greatly feared and often put to death. It was believed that if you saw a witch, her cat would somehow "steal" or control your tongue so you couldn't report the sighting.
Theory #4
This one comes from Evan Morris, the Word Detective: "There's no particular logic to 'cat got your tongue,' except that cats have served as the object of human myth and metaphor for thousands of years."
This list is by no means exhaustive, and there are probably at least a half a dozen more possible explanations. Let's just leave it at this: "Like the history of the word 'cat' itself, the origins of this expression is as mysterious as the Sphinx."