When this definition came up in the (one and only) measure theory course I took as a student, the instructor (Peter Constantin) had this to say about it:
"It says that a set is measurable if you can make change with it."
This explanation has stuck in my mind for the last 15 years, but it is possible that I have remembered it in part because I was never really sure I understood what he meant. Anyway, it sounds good, and if I ever teach a measure theory course (I shudder to imagine the apocalyptic scenario that would necessitate my being called upon to do this: will there be any other mathematicians at all? what color will the sky be?) I might pass it along to my students.