Why does a transcript which is indistinguishable from a fake one show zero knowledge? Is there an intuitive way to understand why having a transcript that the verifier could have created themselves and is indistinguishable from a real transcript , shows zero knowledge?
The information that the verifier gets is all contained in the real transcript. On the other hand, the verifier generates the simulated or fake transcript without interacting with the prover, so all information contained in such a fake transcript is nothing beyond the given fact that "the statement to be proved is true". Now if the real transcript is indistinguishable from the fake one for any adversary with reasonable resources, then clearly these two transcripts contain the same amount of information. Therefore, the real transcript does not leak any information beyond the given fact, i.e., it is zero-knowledge.