One way is to use a combinatorial definition of the derivative. Let $A(z) = \sum a_n z^n$ be a power series. In combinatorics, where $A$ is likely to be an ordinary generating function, $a_n$ is likely to count the number of possible combinatorial structures of some kind on an ordered set with $n$ elements. For example when $a_n = 1$ the structure is "being an ordered set," when $a_n = a^n$ the structure is "being an ordered set together with a choice, for each element of the order, of a letter from an alphabet of size $a$," and so forth.

Then a combinatorial definition of the derivative $A'(z) = \sum na_n z^{n-1}$ is as follows: the derivative is an operation that given an $A$-structure produces a new type of structure, an $A'$-structure. An $A'$-structure "being an ordered set together with an extra element, and an $A$-structure on the new order." This is because for an ordered set with $n$ elements there are $n+1$ possibilities for the extra element and $a_{n+1}$ possibilities for the $A$-structure on the new order.

Then after applying the derivative $k$ times we have added $k$ new elements. If we start from the empty set then there are $k!$ ways to do this, and the set of such ways to do this is naturally a torsor for the symmetric group $S_k$.

One nice property of this definition is that it offers a conceptual interpretation of the Leibniz rule. First, recall that if $A, B$ are two generating functions for $A$-structures and $B$-structures, then an $AB$-structure is a partition of an ordered set into a first segment and a second segment together with an $A$-structure on the first segment and a $B$-structure on the second segment, and the corresponding generating function is the product $AB$. Now, the derivative $(AB)'$ counts the number of ways to add one element to an ordered set and then put an $AB$-structure on the new order. The new element may be either in the first segment or the second segment, and this gives the two terms $A B'$ and $A' B$ in the Leibniz rule.

One way is to use a combinatorial definition of the derivative. Let $A(z) = \sum a_n z^n$ be a power series. In combinatorics, where $A$ is likely to be an ordinary generating function, $a_n$ is likely to count the number of possible combinatorial structures of some kind on an ordered set with $n$ elements. For example when $a_n = 1$ the structure is "being an ordered set," when $a_n = a^n$ the structure is "being an ordered set together with a choice, for each element of the order, of a letter from an alphabet of size $a$," and so forth.

Then a combinatorial definition of the derivative $A'(z) = \sum na_n z^{n-1}$ is as follows: the derivative is an operation that given an $A$-structure produces a new type of structure, an $A'$-structure. An $A'$-structure is "being an ordered set together with an extra element, and an $A$-structure on the new order." This is because for an ordered set with $n$ elements there are $n+1$ possibilities for the extra element and $a_{n+1}$ possibilities for the $A$-structure on the new order.

Then after applying the derivative $k$ times we have added $k$ new elements. If we start from the empty set then there are $k!$ ways to do this, and the set of such ways to do this is naturally a torsor for the symmetric group $S_k$. (A simple example of one benefit of categorification: when you replace numbers by sets, they can support more complicated structures such as group actions.)

One nice property of this definition is that it offers a conceptual interpretation of the Leibniz rule. First, recall that if $A, B$ are two generating functions for $A$-structures and $B$-structures, then an $AB$-structure is a partition of an ordered set into a first segment and a second segment together with an $A$-structure on the first segment and a $B$-structure on the second segment, and the corresponding generating function is the product $AB$. Now, the derivative $(AB)'$ counts the number of ways to add one element to an ordered set and then put an $AB$-structure on the new order. The new element may be either in the first segment or the second segment, and this gives the two terms $A B'$ and $A' B$ in the Leibniz rule.

There is a second, possibly more satisfying, explanation using the theory of combinatorial species and groupoid cardinality. I seem to recall that this explanation was written up by John Baez somewhere in This Week's Finds, but I can't currently find it.