Following more or less directly from this:

- If you walk backwards from where the non-deterministic Turing machine halted, the list of taken state machine transitions is polynomial in length (naturally, as the machine stopped in polynomial time).

- Walking that polynomial length list of actually-taken edges forward through the Turing machine constitutes a polynomial time verification of the solution.

This is the essence of the equivalence between being able to solve problems in NP in polynomial time on a (hypothetical) non-deterministic Turing machine and being able to deterministically verify a solution to those problems in polynomial time.