Hi BrowserUk,
I'm a lackluster mathematics student, but perhaps I can volunteer an explanation that doesn't stray too far from the rigor.
First, I think I would like to recast your statement as "For any set of n elements the number of ways to partition the set into k nonempty subsets is denoted S(n,k) and these numbers are known as Stirling numbers of the second kind."
 S(0,0) = 1 : There is only one way to make no partitions out of nothing.
 S(n,0) = 0 if n>0 : If you have a set of something, you cannot have a partition with no subsetsthe elements of the nonempty set have to go somewhere.
 S(n,1) = S(n,n) = 1 : There is only one way to partition a nonempty set into one nonempty subsetthe subset being the set itself. There is only one way to partition a set of n elements into n nonempty subsetseach element has to be in its own subset (a singleton set).
 S(n,k) = S(n1, k1) + kS(n1,k) : Every partition of a set of n elements into k nonempty subsets can be created from one of the partitions of a set of n1 elements. The sum comes about by looking at the two ways a new, identified, element can be added to the partitions (thus making a new partition that is of a larger set of n elements)
 You can add the singleton set containing just the new element to any partition of n1 elements into k1 subsets, thus creating a partition of a set of n elements into k subsets. There are S(n1, k1) partitions that we can do this to.
 You can add the new element to any nonempty subset of a partition of n1 elements into k nonempty subsets. There are S(n1, k) partitions that we can do this to and since we have k subsets in any partition, we have k ways to modify each of the S(n1, k) partitions, i.e., there are k*S(n1, k) ways to get to a partition of a set of n elements into k nonempty subsets this way.
Note that because we are talking about an identified element being included, then this sum counts each possible partition of a set of n elements into k nonempty subsets exactly once. The identified element must be in any partition of this set of n elements. The identified element is either in a subset by itself (case 1) or in a subset with at least one other element (case 2).
Sometimes the rule S(n,k) = 0 if k>n is given for completeness, i.e., you can't partition something up into more nonempty subsets than for which it has elements, but this is considered obvious and you won't run into the need for it if you start with any reasonable case where n>=k.

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