Variant of what other people already said, but I find the following statement quite intuitive:
- If G is a group and $f: X \to Y$ is a $G$-equivariant map inducing a bijection on orbit sets $X/G \to Y/G$ and isomorphism of all stabilizer groups $G_x \to G_{f(x)}$, then $f$ must be a bijection.
Now in the setting of the 5-lemma, specialize to $X = A_3$, $Y = B_3$, $G = A_2 \cong B_2$ acting by addition on $X$ and $Y$. Then all maps of stabilizers are identified with the isomorphism $A_1 \cong B_1$ in the diagram, and the map of orbit sets is identified with the isomorphism $\mathrm{Ker}(A_4 \to A_5) \cong \mathrm{Ker}(B_4 \to B_5)$.
From this point of view it is also clear that "abelian group" is stronger than necessary. For example, the right-most square need only be given in the category of pointed sets.