I was just learning about the Taylor series recently. It's not too difficult to figure out how to use it on polynomials and what you use it to look for (e.g. tangent lines to functions or calculating natural logs) but I have no idea why it works or how factorials relate to polynomials. I take it that's what you mean when you talk about a layman's explanation? Actually explaining what the thing is instead of just how to use it?