In a way, it doesn't matter if someone means stay out of my business or I'm sincerely grateful that you care. “Thanks for your concern” is a polite way to close a personal conversation. If you hear this phrase, the right course of action is always to drop the topic and move on.
If you are concerned that your interest may have been offensive to them, you can always add something like “Well, I'm available anytime you want to talk, but I won't pry.” This lets the person know that you are sincere, but you also respect their personal privacy.