Here is a sketch of how the skein relations appear in the approach to knot invariants based on braided monoidal categories coming e.g. from representations of quantum groups.

Suppose $V$ is a dualizable object in a braided monoidal category $(C, \otimes)$. This means, among other things, that $V^{\otimes n}$ acquires an action of the braid group $B_n$, and moreover we can meaningfully take traces of the action of a braid in a way which produces an invariant of the knot obtained by closing the braid up. Famously, the Jones polynomial arises in this way when $C$ is taken to be the braided monoidal category of representations of the quantum group $U_q(\mathfrak{sl}_2)$ and $V$ is taken to be the standard $2$-dimensional representation.

Now, one way to describe the skein relation satisfied by the Jones polynomial is that it is a linear relation between three endomorphisms of $V^{\otimes 2}$, namely

• two parallel strands, describing the identity,
• a crossing, describing the braiding $b_{V, V} : V^{\otimes 2} \to V^{\otimes 2}$, and
• the other crossing, describing the inverse $b_{V, V}^{-1}$.

Now, why might there be a linear relation between these three elements? Certainly a sufficient condition is if $\text{End}(V^{\otimes 2})$ is at most $2$-dimensional. And this actually happens in the Jones polynomial case. (The point is that the representation theory of $U_q(\mathfrak{sl}_2)$ is sufficiently similar to that of $\mathfrak{sl}_2$ that it remains true that $V^{\otimes 2}$ is a direct sum of two nonisomorphic irreducibles.)

To my mind, the real problem with trying to motivate skein relations out of thin air is not that you might not be able to simplify knots with them but that there's no reason a priori to expect that a function on knot diagrams defined by repeatedly applying a skein relation should actually be a knot invariant! The point of the categorical machinery of dualizable objects in braided monoidal categories is that you're guaranteed for abstract categorical / topological reasons that you get a knot invariant, and then your job becomes writing down examples in this language. The payoff is that you get much more than a knot invariant, e.g. you also get compatible representations of braid groups.

Here is a sketch of how the skein relations appear in the approach to knot invariants based on braided monoidal categories coming e.g. from representations of quantum groups.

Suppose $V$ is a dualizable object in a braided monoidal category $(C, \otimes)$. This means, among other things, that $V^{\otimes n}$ acquires an action of the braid group $B_n$, and moreover we can meaningfully take traces of the action of a braid in a way which produces an invariant of the knot obtained by closing the braid up. Famously, the Jones polynomial arises in this way when $C$ is taken to be the braided monoidal category of representations of the quantum group $U_q(\mathfrak{sl}_2)$ and $V$ is taken to be the standard $2$-dimensional representation.

Now, one way to describe the skein relation satisfied by the Jones polynomial is that it is a linear relation between three endomorphisms of $V^{\otimes 2}$, namely

• two parallel strands, describing the identity,
• a crossing, describing the braiding $b_{V, V} : V^{\otimes 2} \to V^{\otimes 2}$, and
• the other crossing, describing the inverse $b_{V, V}^{-1}$.

Now, why might there be a linear relation between these three elements? Certainly a sufficient condition is if $\text{End}(V^{\otimes 2})$ is at most $2$-dimensional. And this actually happens in the Jones polynomial case. (The point is that the representation theory of $U_q(\mathfrak{sl}_2)$ is sufficiently similar to that of $\mathfrak{sl}_2$ that it remains true that $V^{\otimes 2}$ is a direct sum of two nonisomorphic irreducibles.)