In the BHK interpretation of constructive logic, a proposition is "true" when you provide a witness for it. You get different notions of constructive logic by defining "witness" in different ways. For example, you could say a witness is a Turing machine that will compute the relevant evidence. For our purposes, let's say for each proposition we assign a set of witnesses. We don't do this arbitrarily though. For an proposition variable, we may assign a set of witnesses arbitrarily, but, in particular, for $A\to B$, we assign as the set of witnesses the function space between the set of witnesses of $A$ and the set of witnesses of $B$. All this is to say, that for this example, a witness of an implication is a (set-theoretic) function from a set of witnesses of $A$ to a set of witnesses for $B$. We also assign the empty set of witnesses to $\bot$, the nullary connective for falsity corresponding to a contradiction1.
For the particular notion of "witness" just described, the principle of explosion is "true" constructively because it is witnessed. In particular, it is witnessed by the empty function. For most of the other common choices for the notion of "witness", there is some analogue of the empty function. One take on generalizing the BHK interpretation, in the case of Intuitionistic Propositional Logic (IPL), is saying that we have a Heyting category of witnesses and the BHK interpretation is a Heyting functor from the (thin) syntactic Heyting category describing IPL to that Heyting category of witnesses. Heyting categories have initial objects which represent $\bot$ and the unique arrow from an initial object to any other object is the generalization of the empty function. Whenever our category of witnesses has an initial object (that's well-behaved in a certain sense and we also have some other "structural" stuff), then we can interpret the $\bot$ connective. Having the $\bot$ connective means having the principle of explosion in the form $\bot\vdash\varphi$ for all $\varphi$.
Of course, we could just drop $\bot$ and weaken the requirement that the category of witnesses be Heyting and that we have a Heyting functor to not include the initial object stuff. That's a completely reasonable thing to do and will lead to a paraconsistent logic such as minimal logic. It's not so much "do we support the principle of explosion" as "do we have (something equivalent to) the $\bot$ connective". However, dropping $\bot$ means we need a different story for negation, $\neg$, or dropping we need to drop negation entirely. For intuitionistic logics, negation is usually defined as $\neg\varphi\equiv\varphi\to\bot$.
1 I strongly recommend providing $\bot$ as a primitive connective (for logics where contradictions make sense). Always talking about $P\land\neg P$ is tedious and kind of hacky. Why do I need to arbitrarily choose some irrelevant proposition $P$ just to talk about a fundamental concept like contradiction?