I understand that in intuitionistic logic, a statement $A \to B$ is meant to be interpreted as "from a proof of $A$ we can construct a proof of $B$."

However, I'm confused by the following axiom:

$$\neg A \to (A\to B).$$

This seems to mean "from a proof of $\neg A$ we can construct a proof that if we were given a proof of $A$ we could construct a proof of $B$."

But why should that be true? What is the justification for including this axiom?

I guess an argument would from the principle of explosion: "if we have proven $\neg A$ then the only possible way we can also be given a proof of $A$ is if our system is inconsistent, and therefore any $B$ can be proven." But this seems somehow odd to me. I'm under the impression (perhaps wrongly?) that in classical logic the principle of explosion is a consequence of the law of the excluded middle, so if we're rejecting the LEM, why would we want to keep the principle of explosion?

I know that minimal logic exists, and from what I can tell it seems to essentially be intuitionistic logic without this axiom, so it seems to possible to reject both the LEM and the principle of explosion. I'm also aware that there are some practical proofs that work in intuitionistic logic but not in minimal logic. However my question is more on the philosophical side: for the intuitionists, what was the justification for the $\neg A \to (A\to B)$ axiom?