My thoughts:
Consider an array of $N$ characters, $C$, such that C= [$c_1$, $c_2$, ..., $c_N$].
Now, the structure of a $K$-partition of $C$ can be represented as a corresponding array of $K$ characters,
$P$ = [$(1..K)_1$, $(1...K)_2$, ..., $(1...K)_N$]
where (1...K) denotes a possible index for the element of $C$ in the partition of $C$.
For example, if $C$ = [a,b,c,d,e], then the partition $\{(a,b), (c), (d,e)\}$ corresponds to $P$ = $[0,0,1,2,2]$
Since, by definition of a partition, every element of a $K$-partition must be non-empty, every possible index of a $K$-partition of must be represented at least once in the array of characters $P$ = [$(1..K)_1$, $(1...K)_2$, ..., $(1...K)_N$]. Since every partition of $C$ can be represented as a corresponding $P$, $S_2(N,K)$ denotes the number of possible $P$ such that every possible index in the set {1,2,...,K} is represented at least once.
$K^N$, on the other hand, counts all $P$ where indexes of a $K$-partition may or may not be represented. That's to say, $K^N \geq S_2(N,K)$
EDIT: Furthermore, if we are interested in counting all partitions of $C$ and all possible orders of elements (subsets of $C$, not the elements of $C$ themselves) in the partitions, then I believe $K^N \geq K! \cdot S_2(N,K)$ since our count of strings with $K^N$ possibilities does not treat equal-ordered partitions as equivalent.
Is there a case I'm missing? If not, is my understanding of why the claim is true flawed?
Otherwise, I'm aware that $K^N$ is not a very good upper-bound in the sense that it's very conservative, but I'm in a context where simplicity of the expression is very desirable.
Yes, it is indeed true that $S_2(n,k)\le {k^n}/{k!}$. Put another way, $k!S_2(n,k)$ is the number of ways to place $n$ distinct objects into $k$ distinct boxes so that no box is empty, while $k^n$ counts all the ways to place $n$ objects into $k$ distinct boxes.