I have only heard this as 'I sort of got to thinking', but it's very colloquial.
It's a way of expressing that you've been pondering over something, especially when reflecting upon how to solve a tricky problem or difficult situation. Often it's used when you come up with a good idea or new approach to something.
If you have a question about the English language and would like to ask one of our many English teachers and language experts, please click the button below to let us know: