Since long exact sequences come from splicing together short exact sequences, you might as well worry about the case where $A_1=A_5=B_1=B_5=0$ (at least as far as intuition is concerned). This follows from the Snake Lemma, of course, but the version where the outside vertical maps are isomorphisms is an even easier diagram chase. Perhaps that will illuminate.
In terms of why it's true without chasing elements, think about the same simplified version, but just for Abelian groups. In general, of course, you can have $G/H \cong G'/H'$ for lots of groups $G,G'$ and respective subgroups $H,H'$. In general, that isomorphism won't even lift to a homomorphism $G \to G'$, much less an isomorphism. Similarly, you could have $H \cong H'$ without that isomorphism extending to a homomorphism $G \to G'$. If, however, you have both, then the attempt to lift the one isomorphism and the attempt to extend the other both succeed.
