The number of perfect matchings corresponds to the Matrix Permanent 
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?
 A: 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.
