I am teaching an advanced undergraduate class on topology. We are doing introductory knot theory at the moment. One of my students asked how do we know to use this skein relation to compute all these wonderful polynomial invariants of knots.

I was not trained as a knot theorist, so I was at loss. Intuitively it does not seem to be that powerful, because it is not clear (to me at least) that you can always pick a recursive relation involving simpler knots. Could somebody help me motivating my students here? In other words,

Why are the skein relations so useful for computing polynomial invariants of knots, and when did people realize that is the case?

Thanks.