What is the practical use of Laplace Transform?
The "s" that you are referring to is not the "s" of the Laplace Domain. It is called "slip" and the definition is

\$s = \frac{n_R}{n_S} = \frac{n_S-n}{n_S} = 1 - \frac{n}{n_S}\$

where

\$n\$ is the rotational speed of the rotor

\$n_S = \frac{f_S}{p}\$ is synchronous rotational speed

\$n_R = n_S - n\$ is the relative rotational speed

\$f_S\$ is the electrical frequency - typically 50 or 60 Hz

and finally

\$p\$ are the # of poles of the stator

So, slip is a measure of how away the rotor speed is from the synchronous speed.

Any textbook provides the proof of how mechanical power can be represented by the power that is consumed by a resistor, the value of which is  \$R_R\frac{1-s}{s}\$



Laplace Domain is basically a tool for solving the ODEs, by make them linear equations - it also gives information for the spectral content of the system, but that is a different story.