Understanding forking lemma and oracle replay attack
We use the fact that forger $\mathcal{F}$ is nothing else but a Turing machine with some random tape. And since $\mathcal{F}$ is in our possession, we can choose the tape.
Therefore, we can rerun $\mathcal{F}$ with the same tape, and behavior of $\mathcal{F}$ will be just the same (until the point where the oracle will return different value and desired "forking" occurs).