Imagine having a where's waldo book, looking for waldo with a friend, and your friend says he's found him... you don't believe him, but he won't show it to you, because then you'll know the location of waldo too. How can he prove, that he found it, without showing the location to you? By taking this punctured-foil approach, he can prove it to you, that he actually did find it, and do it in a way, where you know he really found it, but you still don't know where it is.

Moving the foil randomly enough might work with a book, but this is just an example of a mathematical principle. An rsa key might be between 1 and 2^4096, randomly guessing the number is practically impossible (atleast with current computing power and without waiting literally millions of years). But if the owner uses the key and signs a number you gave him, you verify the signature with his public key, you know that he actually has the private key, but you dont know the actual key.