Theorem 6.52 in Lee's Introduction to Smooth Manifolds, 2nd ed., says
Suppose $M$ and $N$ are smoothe manifolds and $S \subset M \times N$ is an immersed submanifold. Let $\pi_M$ and $\pi_N$ denote the projections from $M \times N$ to $M$ and $N$, respectively. The following are equivalent:
- $S$ is the graph of a smooth map $f: M \to N$.
- $\left. \pi_M \right|_S$ is a diffeomorphism from $S$ onto $M$.
- For each $p \in M$, the submanifolds $S$ and $\{p\} \times N$ intersect transversely in exactly one point.
The proof is left as an exercise. I'm comfortable with 1 $\iff$ 2 and 1 $\implies$ 3. In going from 3 to 2, perhaps you can tell me if this is correct, or if I can trim the fat from this argument:
Let $i:S \to M \times N$ be the immersion. By 3 we know that $\pi_M \circ i$ is a smooth bijection. Therefore the dimension of $S$ must equal the dimension of $M$. To show it is a diffeomorphism, it suffices to show that the derivative is nonsingular at each point, since then we have a injective local diffeomorphism. To show that $D(\pi_M \circ i)_s$ is invertible, it suffices to show it is surjective. By transversality we have $T_{i(s)}(i(S)) + \{0\} \times T_{\pi_N \circ i(s)}N = T_{i(s)}(M \times N)$. Then, taking $(v,w) \in T_{i(s)}(M \times N)$, it is clear that $v$ is in the range of $D(\pi_M \circ i)_s$.