I am someone with "deep knowledge", but HN is not the proper place for this discussion. See
https://people.cs.georgetown.edu/jthaler/ProofsArgsAndZK.htm... for the gory details.
Here is a hopefully simple example of how this ZKP thing may even be possible. Imagine that you give me a Sudoku puzzle. I solve it, and then I want to prove to you that I have solved it without telling you the solution. It sounds impossible, but here is one way to do it.
I compute the solution. I randomly scramble the digits 1-9 and I put the scrambled solution in a 9x9 array of lock boxes on a table. I have the keys to the 81 locks but I am not giving you the key yet. You randomly ask me to open either 1) one random row chosen by you; 2) one random column chosen by you; 3) one random 3x3 block chosen by you; or 4) the cells corresponding to the original puzzle you posed to me. In total you have 28 possibilities, and assume that you choose them with equal probability. You tell me what you want and I open the corresponding lockboxes. You verify that the opened lock boxes are consistent with me knowing a solution, e.g. all numbers in a row are distinct, the 3x3 block consists of distinct numbers, etc. If I am cheating, then at least one of your 28 choices will be inconsistent, and you catch me with probability 1/28, so if we repeat this game 1000 times, and I don't know the solution, you will catch me with probability at least 1-(1/28)^1000 which is effectively 1. However, every time we repeat the game, I pick a different random scrambling of the integers 1-9, so you don't learn anything about the solution.
All of ZKP is a fancy way to 1) encode arbitrary computations in this sort of protocol, and 2) amplify the probability of success via clever error-correction tricks.
The other thing you need to know is that the protocol I described requires interaction (I lock the boxes and you tell me which ones to open), but there is a way to remove the interaction. Observe that in the Sudoku game above, all you are doing is flipping random coins and sending them to me. Of course you cannot let me pick the random coins, but if we agree that the random coins are just the SHA256 hash of what I told you, or something else similarly unpredictable, then you will be convinced of the proof even if the "coins" are something that I compute myself by using SHA256. This is called the "Fiat-Shamir transformation".
How do we implement the lock boxes? I tell you SHA256(NONCE, VALUE) where the NONCE is chosen by me. Given the hash you cannot compute VALUE. To open the lock box, I tell you NONCE and VALUE, which you believe under the assumption that I cannot find a collision in SHA256.