This is almost always said in response to someone. If you are discussing, and your opinion is set, but the other person tells you something you had not considered before, and it may possibly change your opinion, as you express that you may change your opinion, you may start with, "You got me thinking, now ... " This is a colloqial expression. A more proper expression may be, "You have me thinking ... " Other expressions might include: Now that I know that, I'm thinking ... I hadn't considered that, but ...

It's a minor point, but in spoken British English, at least, a more correct form would be: 'You've got me thinking now, ....' I know that we can argue till the cows come home (a nice idiom meaning 'endlessly' !) about the relative 'correctness' of 'got' 'have' and 'have got' , but I think the form here is meaningful 'Have got' renders the idea of something which the other person has just done with respect to the speaker's attitudes and way of thinking. It's another way of saying 'What you have just said has made me reconsider my opinion on that topic'. And, as Steve says, this phrase is a reaction to something which another person has said, and would be followed by the speaker saying something like, 'Hmm, maybe we were wrong after all when we ..... etc '

you got/let/make me thinking now