Warning: Oversimplified.
Most, if not all interactive Zero-Knowledge proofs rely on the randomness of the verifier. Taking the Ali Baba cave example, since Alice cannot predict what path Bob wants her to come out of, it is statistically improbable that she will be able to fake the proof, especially when this "experiment" is done multiple times.
In case you don't know, a hash funtion is a pseudo-random function. This means that, given the same input, it will always generate the same output. A cryptographic hash function is a one-way function, one where it is infeasible to create a preimage attack, or simply put, finding X in H(X)=Y, given only Y.
The Fiat-Shamir heuristic simply replaces the verifier's randomness (and therefore the interaction) with a random oracle, or in concrete terms, a cryptographic hash function. Since the prover cannot predict the output of a cryptographic hash function, this provides the randomness necessary for the scheme to work.
In that case, Alice wants to prove that she can unlock the door in a cave C. She goes inside the cave, generates a hash H(C), and depending on the output, goes in either path A or B and comes out the other path. Bob, without ever telling Alice anything, simply checks that Alice went through the correct path (if H(C)=A or H(C)=B) and that she exited through the other path.
You could adapt this to make it harder to fake, by computing H(C||n) instead of H(C), where n is a number starting from 0 that increases by 1 every time the experiment is conducted, and running the experiment multiple times.
This Wikipedia article is a really good one and gives a more concrete example.
