It's an interesting question: how those who work on quantum algorithms think about quantum algorithms. As someone who worked on quantum algorithms for many years, the interesting intuition that I developed was not what made quantum algorithms tick, but more along the lines of "when is a method likely to NOT yield a quantum speedup." That is I found I could pretty quickly tell when I was doing something that had a nice classical analogy and was therefore doomed to failure. Classic examples where when your intuition about why you might get a speedup involved exploring an exponentially large solution space without some reason that interference could be used to help dig out the interesting parts of the global superposition you created.

An interesting example of how one discovers a quantum algorithm is the story of the NAND tree algorithm (See http://www.scottaaronson.com/blog/?p=207 for a nice explanation.) Quite literally the quantum algorithm was discovered by thinking about interference of particles hoping around, combined with physicists uncanny ability to calculate scattering cross sections. Crazy stuff!