A question about verifier in zero knowledge proof
The meaning of this has two aspects:


After running the protocol successfully, the verifier is convinced (with overwhelming probability) that the information is genuine. 
The verifier can by no means extract the information from the protocol. It is simply impossible (not unlikely, actually impossible).


Very simplified a zero knowledge protocol goes like this:


First, the prover sends some information to the verifier
Then the verifier generates a random value (often just a single bit) and sends that to the prover
The prover answers based on that random value, and the verifier can check that answer combined with the first value and his random value.


The protocol is set up such a way, that only if the prover truly knows the secret information, he can send the correct answer in the last step with certainty. Otherwise he has to rely on luck. So if the verifier used random values in the second step and the prover didn't know them in the first step, there is a certain chance (often probability $0.5$) that the verification in the end does not work out.

And lastly, the above steps are done multiple times and only if the prover passed all of them the verifier is convinced. With probability $1/2$ per round, the remaining uncertainty after $k$ rounds is $\frac{1}{2^k}$.