Conway was the person who gave us the concept of a skein relation (although, in retrospect, a skein relation does appear in the original paper of Alexander on the Alexander polynomial). What we are discussing here are linear skein relations (Conway considered also non-linear skein relations, although he never published on the subject, and the idea seems to have been virtually forgotten). Conway's idea, as I luckily overheard him explain it to Cameron Gordon, was to consider knot invariants not as maps of the set of knots to the set of polynomials (for instance), but as maps from some sort of space of knots, locally characterized by how they behave on knots in "close proximity". What does it mean for two knots to be close to each other? For Conway it meant that they differ by some simple local tangle replacement. That is the skein.
As was mentioned in other answers, this dovetails nicely with later ideas of TQFT and quantum knot invariants, one of whose defining properties is that they are determined locally on small pieces of the knot or 3-manifold, which are then glued together. It's frightening just how close Conway was to discovering the Jones polynomial using his approach- we can only dream how that might have changed the history of knot theory.
I agree with all the other answers, therefore, that the motivation for skein relations is philosophical rather than as a tool to calculate, for which it is notoriously unsuited.

Conway was the person who gave us the concept of a skein relation (although, in retrospect, a skein relation does appear in the original paper of Alexander on the Alexander polynomial). What we are discussing here are linear skein relations (Conway considered also non-linear skein relations, although he never published on the subject, and the idea seems to have been virtually forgotten). Conway's idea, as I luckily overheard him explain it to Cameron Gordon, was to consider knot invariants not as maps of the set of knots to the set of polynomials (for instance), but as maps from some sort of space of knots, locally characterized by how they behave on knots in "close proximity". What does it mean for two knots to be close to each other? For Conway it meant that they differ by some simple local tangle replacement. That is the skein.
As was mentioned in other answers, this dovetails nicely with later ideas of TQFT and quantum knot invariants, one of whose defining properties is that they are determined locally on small pieces of the knot or 3-manifold, which are then glued together. It's frightening just how close Conway was to discovering the Jones polynomial using his approach- we can only dream how that might have changed the history of knot theory.
I agree with all the other answers, therefore, that the motivation for skein relations is philosophical rather than as a tool to calculate, for which it is notoriously ill-suited.