How can you kindly remind people when they are being arrogant?

The best course of action is silence. If nobody reacts to the person's bragging, then eventually, s/he will stop. Bragging is akin to acting, and it's not acting if there's no audience. What happens to comedians whose audiences never laugh? They never become professional.

I would add to the above that it is not your job to change other people, and you will frustrate yourself no end if you try. The only way people change is to change themselves, and they have to have a reason.

As far as directly answering the question, which is what we do here on WikiAnswers, you could -- if you must -- remark, "You know, that seems a little bit arrogant to me. Would you explain exactly where you are coming from on that?"