The Wikipedia answer is one answer that is commonly used: replace sets with categories, replace functions with functors, and replace identities among functions with natural transformations (or isomorphisms) among functors. One hopes for newer deeper results along the way.
In the case of work of Lauda and Khovanov, they often start with an algebra (for example ${\bf C}[x]$ with operators $d (x^n)= n x^{n-1}$ and $x \cdot x^n = x^{n+1}$ subject to the relation $d \circ x = x \circ d +1$) and replace this with a category of projective $R$-modules and functors defined thereupon in such a way that the associated Grothendieck group is isomorphic to the original algebra.
Khovanov's categorification of the Jones polynomial can be thought of in a different way even though, from his point of view, there is a central motivating idea between this paragraph and the preceding one. The Khovanov homology of a knot constructs from the set of $2^n$ Kauffman bracket smoothing of the diagram ($n$ is the crossing number) a homology theory whose graded Euler characteristic is the Jones polynomial. In this case, we can think of taking a polynomial formula and replacing it with a formula that inter-relates certain homology groups.
Crane's original motivation was to define a Hopf category (which he did) as a generalization of a Hopf algebra in order to use this to define invariants of $4$-dimensional manifolds. The story gets a little complicated here, but goes roughly like this. Frobenius algebras give invariants of surfaces via TQFTs. More precisely, a TQFT on the $(1+1)$ cobordism category (e.g. three circles connected by a pair of pants) gives a Frobenius algebra. Hopf algebras give invariants of 3-manifolds. What algebraic structure gives rise to a $4$-dimensional manifold invariant, or a $4$-dimensional TQFT? Crane showed that a Hopf category was the underlying structure.
So a goal from Crane's point of view, would be to construct interesting examples of Hopf categories. Similarly, in my question below, a goal is to give interesting examples of braided monoidal 2-categories with duals.
In the last sense of categorification, we start from a category in which certain equalities hold. For example, a braided monoidal category has a set of axioms that mimic the braid relations. Then we replace those equalities by
$2$-morphisms that are isomorphisms and that satisfy certain coherence conditions. The resulting $2$-category may be structurally similar to another known entity. In this case, $2$-functors (objects to objects, morphisms to morphisms, and $2$-morphisms to $2$-morphisms in which equalities are preserved) can be shown to give invariants.
The most important categorifications in terms of applications to date are
(in my own opinion) the Khovanov homology, Oszvath-Szabo's invariants of knots, and Crane's original insight. The former two items are important since they are giving new and interesting results.
Addressing just the primary question, I think there are clear examples
where English-speaking mathematicians have been slow to catch up with
developments well-known to German or Russian speakers. One I would
mention is the result that surface mappings are generated by twists,
published by Dehn in 1938, but rediscovered by Lickorish in 1963. The
big advances in the theory of surface mappings due to Thurston in the
1970s were, I think, somewhat slowed by the fact that he had to
rediscover many results of Nielsen published in German or Danish in
the 1920s.
At a more trifling level, I was once embarrassed to learn that a result
I published in 1987 was well-known to Russian mathematicians.
Best Answer
I'm not an expert on the sheaf-theoretic approach to probability theory, but a quick look at the paper you're asking about shows that it's a 6 page conference proceeding from 2017 that defines a new notion and reads like a set of lecture notes. I think it's a bit early to be asking for big applications to probability theory. That said, if you use Google Scholar to search for who cites Alex Simpson's paper, you find two. The first is about constructive measure theory, so should be of interest to logicians and others wanting a firmer foundation for probability theory. The second is a PhD thesis that goes much more in depth than Simpson's paper about the properties of "probability sheaves" and investigates connections to topos theory.
From the abstract: "In this dissertation, we emphasize how sheaves and monads are important tools for thinking about modern statistical computing." The abstract goes on to advertise applications to hypothesis testing and the analysis of data sets with missing data, pretty important topics. Chapter 3 contains lots of history of previous attempts to bring probability theory under the umbrella of category theory, and also includes applications to probabilistic programming (whatever that is). If you're interested in this field, I think you will want to read these references (plus the blog post by Tao that Simpson cites), and you may need to give it time before super compelling applications arrive.