There are two possible meanings for the sentence "f : M → N admits local sections",
so let's first disambiguate.
Meaning 1: For every point of N, there exists a neighborhood of that points and a section from that neighborhood back to M.
That's what people typically check in order to verify that, say, a map is a $G$-principal bundle.
Meaning 2: For every point m ∈ M, there exists a neighborhood of $f(m)$, and a section s from that neighborhood back to M, subject to the extra condition that $s(f(m))=m$.
Clearly, you care about the second meaning of that sentence.
It is correct that a map is a submersion (not necessarily surjective!) iff it admits local sections.
If a map has local sections, then the maps on tangent spaces are sujective: that's just obvious.
Conversely, if a map is surjective at the level of tangent spaces, you first pick a local section of the maps of tangent spaces. Then, to finish the argument, you use the fact that
any subspace of the tangent space $T_mM$ is the tangent space of a submanifold of M, and apply the implicit function theorem.
Note: if you care about infinite dimensional Banach manifolds, then the existence of a section for the map to tangent spaces needs to be assumed a separate condition. Indeed, it's not enough to assume that the map of tangent spaces is surjective, since it's not true that any surjective map of Banach spaces has a section.
Note: For complex varieties, you don't have the implicit function theorem, so it doens't work. Counterexample: the map $z\mapsto z^2$ from ℂ* to itself. The fix is to pass the the "étale topology"... but that's another story.