They recover the S and K by applying U to itself in interesting ways, but that's like saying 'this tool is so universal, you can use it as a hammer and a chisel by banging one instance of the tool with another tool just like it'.
That's not a very mathematical way of putting it I guess but I believe that is more or less the spirit of it.
There is also a U combinator referenced on this page:
http://www.ucombinator.org/
as U = λ f . f(f) - haven't had time to sit down and actually work it through though.
[Edit: I notice that page says that U enables universal computation, not that it is sufficient.]