I want to show that the number of perfect matchings in a bipartite graph is precisely the permanent of the adjacency matrix of the graph.
This seems somehow quite natural to me. But how could I give a rigorous proof of this?
I would say that it is enough to note that
$$\Pi_{i=1}^n a_{i,\sigma(i)} = 1$$
if and only if all vertices are matched. The permanent now only counts how often this is the case. Is this enough?
Essentially yes, I'd just specify which matching you mean. One way to say it more formally: "the condition $\prod_{i=1}^n a_{i,\sigma(i)}=1$ is equivalent to saying that $\{i,\sigma(i)\}$ is an edge for each $i$, so a permutation $\sigma$ satisfying it corresponds to the perfect matching that matches $i$ with $\sigma(i)$".
To be excessively hyper-formal, you could first say this product is always either 0 or 1, so the permanent is equal to the number of $n$-permutations $\sigma$ such that $\prod_{i=1}^n a_{i,\sigma(i)}=1$. Then define a bijection between such permutations and perfect matchings (usually the most obvious way is by defining maps $f,g$ in one way and the other and checking $f\circ g=\mathrm{id}$ and $g\circ f=\mathrm{id}$, rather than checking f is injective and surjective). But that's obvious enough from the above description; it should suffice to point somehow, however informally, that you understand what is what: that $i$ corresponds to a vertex, $a_{i,\sigma(i)}=1$ to the existence of an edge, and $\sigma$ to a matching candidate.