Let $G = G_1 \ast G_2$. Then $$G/[G,G] \simeq G_1/[G_1, G_1] \oplus G_2/[G_2,G_2].$$
This is Munkres, section 69, exercise 1; with the hint to use the extension condition for free groups and direct sums in each direction, so find 2 homomorphisms, and show that they are mutual inverses.
It is easy to find some homomorphism in either direction:
(i) left-hand-side (LHS) to right-hand-side (RHS): as RHS is the direct sum of two abelian groups, it is abelian. Any homomorphism from $G$ to RHS will then induce a homomorphism from LHS to RHS (eg, from the elementary first isomorphism theorem).
(ii) RHS to LSH: eg, mapping $x[G_1, G_1]$ to $x[G,G]$ is a homomorphism that extends to one on $G$, using the extension condition for direct sums.
However, (i) is just an abstract statement from which I do not see how to make progress in showing that we have mutual inverses (nor does (ii) help me much so far).
Note: at this point in the book, Munkres has only introducted fundamental groups, covering spaces, deformation retracts, etc, plus a few chapters on free groups and direct sums, to the level of showing the fundamental groups of the Torus, some discussion on figure 8 etc. There are probably very abstract ways in which this might follow; but if you could kindly stick to as elementary a proof or hint as possible, I would appreciate! My group theory knowledge is currently at about the level of one class of group theory.
P.S.: This is not homework. I'm reading Munkres for fun, and try to do every exercise. Usually, this goes fine. But I'm really stuck here.