It has a meaning of I'm ashamed to tell you or I regret what I did.

Thank you for your explanation.

You say, but with some regret. You feel a need to say, as it says, but for some reason, you regret the actual thing that you are about to say. I regret to say that this is the best generic explanation that I have.

The function of "regret to say" is that, now you wish you could have helped them which you did not. You are feeling sorry now that you did not help them.