This format is somewhat common, but it is a little bit "slang." In this situation, "get to" means "start" or "began." Some examples: "Last night, I was feeling fine, and then I got to thinking about my ex-girlfriend, and then I was sad." (past tense) "When I get to thinking about that project, I become stressed and anxious." (present tense - sort of, haha) This is one of those phrases that you don't need to use, but it is good if you can understand it when someone else says it. You can use "start" or "begin" and have the same meaning. For example: "Last night, I was feeling fine, and then I started thinking about my ex-girlfriend, and then I was sad." "Last night, I was feeling fine, and then I began thinking about my ex-girlfriend, and then I was sad." or "When I start thinking about that project, I become stressed and anxious." "When I begin thinking about that project, I become stressed and anxious." Good luck!

I think they mean the same. Basically Started doing something triggered by something. I think it is more US English? Let’s see what others think .

I think the expression you're referring to is: "it got me thinking about it". It basically, "event A was the trigger/reason why I began thinking about something"

I got to thinking….. It is usually triggered by some experience, that got you thinking and you made a conclusion or discovery based on that. The thinking is a noun her not a verb. The economy shut down in 2020, oil was at its lowest, everyone needs oil….so I got to thinking, that it may be a good time to invest.