Why does a surjective map $\pi_1(M)\twoheadrightarrow F$ of the fundamental group of a manifold $M$ onto a free group $F$ over $n$ generators induce a continuous map $M\twoheadrightarrow\bigvee^n S^1$ onto a wedge of $n$ circles?
I've seen it mentioned without proof in various places, such as Jaco, page 3: "Let T denote a wedge...", Leininger, page 3, proof of lemma 2.1, etc.
Is this true for non-orientable manifolds?
Orientability has nothing to do with it; it's true for arbitrary CW complexes. The following more general thing is true: any map $\pi_1(X) \to G$, for $G$ a discrete group, arises from a map $X \to BG$, where $BG$ is the classifying space of the discrete group $G$, or equivalently an Eilenberg-MacLane space $K(G, 1)$. When $G$ is a free group, $BG$ is a wedge of circles.