Where does it come from? Is it universal or specific to a particular region?
The word get (past tense got) can mean begin, especially coupled with to. See here.
get to it
The reversal of to get into get to tends to be an informal usage (when used in this way).
I believe it to be an extension of the concept of get as meaning to reach (a state of). This is a very old usage of the word dating back to geta ca. 1200 in Old Norse.