Non-deterministic algorithms and Tautologies
An algorithm computing $V$ can be thought of as a non-deterministic algorithm which gets only the first parameter $\tau$ and guesses $P$, verifying that $P$ is a proof of $\tau$.

If you've ever seen the definition of NP using witnesses, in this definition $V$ is accepts the language of all tautologies $\tau$ with proofs in a certain proof system. The witness that such a proof exists is the proof itself $P$, and the function $V$ verifies that the proof is valid for $\tau$.

Another equivalent way of defining a proof system is as a polynomial time function which accepts a proof $P$ and outputs a proposition $\tau$. The two properties satisfied by such a system $S$ is (i) for all $P$, $S(P)$ is a tautology, (ii) for every tautology $\tau$ there exists a proof $P$ such that $S(P) = \tau$. It is an easy exercise to convert between the two definitions.

If you want to learn more, look up Cook–Reckhow proof system.