“Thank you very much,” is a phrase often used to end a conversation politely. In this context, the phrase would tell someone that the Dursleys don’t want to discuss the topic any further. You might also hear, “Thank you for your concern,” or “Thank you for asking,” which makes it more explicit what the other person is being thanked for. It’s a formality, and doesn’t mean someone is truly grateful. If whoever says this doesn’t ask for help or advice, that communicates that the person they’re talking to should not offer.
Unlike, for example, “Have a nice day,” this doesn’t necessarily mean the person saying it wants to end the conversation, only change the topic.
So, we’re being asked to imagine that people sometimes wondered or suggested that there was something a little different about the Dursleys. Whenever that happened, they would insist that, no, they were “perfectly normal,” and immediately halt that line of conversation. They might even have said they were “perfectly normal” themselves, without being asked, and then said “Thank you very much,” to tell other people not to suggest otherwise. That would be very strange behavior (suggesting that there really is something not-so-normal about them that they want to hide), but the Dursleys are the kind of slapstick characters who might do that.