I suspect that no very satisfying answer to this question is known, so let me just point out that there are a lot of different things one might mean by a "category-theoretic understanding" of a large-cardinal axiom. I am far from an expert, so I hope that others can help flesh this out and correct me, but here are some themes that I happen to have seen in the literature:

1. Set theorists have responded to the paucity of elementary embeddings $V \to V$ by considering elementary embeddings $V \to M$ where $M$ is not $V$. Whereas category theory suggests studying endofunctors $\mathsf{Set} \to \mathsf{Set}$ which satisfy nice properties weaker than being elementary embeddings (actually the first approach can be viewed as a special case of the second, since $M$ is generally assumed to be a non-elementary substructure of $V$, so we can compose $V \to M \subset V$ to get an endofunctor $V \to V$). This approach is much less developed; the only examples I happen to have come across are at the level of measurable cardinals:

 - As David Roberts alluded to in the comments, [Isbell](https://projecteuclid.org/euclid.ijm/1255456274) essentially showed that a full subcategory of $\mathsf{Set}$ is [codense](http://ncatlab.org/nlab/show/codense+functor) iff it is not bounded by any measurable cardinal. A subcategory $A \subseteq B$ is codense (when $B$ is suitably complete) iff its _codensity monad_ (a certain endofunctor $B \to B$, namely the right Kan extension of the inclusion $A \to B$ along itself) is isomorphic to the identity.  The codensity monad of a full subcategory $A \subseteq \mathsf{Set}$ is the functor $\mathsf{Set} \to \mathsf{Set}$ sending $X$ to the set of $A$-complete ultrafilters on $X$, hence from a category-theoretic perspective, measurable cardinals can be viewed as connected to the study of certain endofunctors $\mathsf{Set} \to \mathsf{Set}$ which are _not_ elementary embeddings.

 - Another relationship between endofunctors of $\mathsf{Set}$ and measurable cardinals is [Blass's](http://msp.org/pjm/1976/63-2/p03.xhtml) result that there is a measurable cardinal if and only if there is a left exact endofunctor of $\mathsf{Set}$ not isomorphic to the identity. Both of these results are related to [Borger's](http://www.sciencedirect.com/science/article/pii/0022404987900417) result that the endofunctor of $\kappa$-complete ultrafilters is terminal among endofunctors of $\mathsf{Set}$ preserving $\kappa$-small coproducts.

2. I'm having a harder time getting a grip on where category-theoretic statements of Vopenka's Principle fit, largely out of ignorance. But it seems like roughly, any ZFC-independent statement about accessible categories is equivalent to Vopenka's Principle, making it hard to see how to generalize the approach. The idea that a large cardinal axiom can be equivalent to a structure axiom about essentially arbitrary classes of first-order models is pretty cool -- I guess this is analogous to having combinatorial statements equivalent to smaller large cardinals. It provides a pretty compelling picture of what Vopenka's Principle _means_. 

3. Type theory has a natural categorical semantics. And it's natural to try to formulate large cardinal axioms in terms of the universes in type theory. There's some work on this by [Palmgren](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.221.1318), but I think it's dealt mostly with smaller large cardinal axioms.