Laplace Relation to Capacitors and Inductors
Most simply 1/s is the Laplace operator for integration, and s, for differentiation. Start with the differential equations describing the reactive circuit elements, like so.

\$ i=C\frac{dV}{dt} \$

\$ V=\frac{1}{C}  \int i dt \$

\$ V = L \frac {di}{dt} \$

Now, try to replace things with the Laplace operators and a simple Ohms Law relationship V=iR and see what the complex impedances must be.

For example, \$ V(s) = \frac{1}{C} \frac{1}{s} I(s) \$

yields an effective impedence of \$ \frac{1}{sC}\$, and it would be a similar derivation for inductance.

It all comes from the relationships between the Laplace operators and the differential equations.  This used to be routinely taught in differential equations classes, and engineering students would eventually fill in the blanks to understand how it applied to what they learn in circuits.  Unfortunately, diffeqs are just not taught with the same rigor anymore.