A recursive function is like a spring you compress a bit on each call. On each step, you put a bit of information (current context) on a stack. When the final step is reached, the spring is released, collecting all values (contexts) at once!
Not sure this metaphor is effective... :-)
Anyway, beyond the classical examples (factorial which is the worst example since it is inefficient and easily flattened, Fibonacci, Hanoi...) which are a bit artificial (I rarely, if ever, use them in real programming cases), it is interesting to see where it is really used.
A very common case is to walk a tree (or a graph, but trees are more common, in general).
For example, a folder hierarchy: to list the files, you iterate on them. If you find a sub-directory, the function listing the files call itself with the new folder as argument. When coming back from listing this new folder (and its sub-folders!), it resumes its context, to the next file (or folder).
Another concrete case is when drawing a hierarchy of GUI components: it is common to have containers, like panes, to hold components which can be panes too, or compound components, etc. The painting routine calls recursively the paint function of each component, which calls the paint function of all the components it holds, etc.
Not sure if I am very clear, but I like to show real world use of teaching material, as it was something I was stumbling upon in the past.
