For a non-mathematical explanation of the core idea behind SNARK, see http://manishearth.github.io/blog/2016/03/05/exploring-zero-... (I've attempted to make it accessible to people with a minimal background in these things)

(I wrote this after being fascinated by the explanation of the core idea in https://people.xiph.org/~greg/simple_verifyable_execution.tx..., but finding that post too complicated to share with my non-crypto physics friends)