When you take the Laplace transform of a function $\displaystyle f(t) $ of the varible $\displaystyle t $, you get a function $\displaystyle F(s) $ of a different variable $\displaystyle s $.
The rule
$\displaystyle {\cal L} f(at) = \frac {1}{a} F(s/a) $
tells you how to find the Laplace transform of $\displaystyle f(at) $ when you only know the Laplace transform of $\displaystyle f(t) $.
For example, given that $\displaystyle {\cal L} \sin(t) = \frac {1}{s^2 + 1} $,
find $\displaystyle {\cal L} \sin(3t) $.
$\displaystyle {\cal L} \sin(3t) = \frac {1}{3} \frac {1}{ (s/3)^2 + 1} $
$\displaystyle = \frac {1} {3} \frac {1}{ \frac {s^2}{ 9} + 1} $
$\displaystyle = \frac {3}{9} \frac {1}{ \frac {s^2}{ 9} + 1} $
$\displaystyle = \frac {3}{s^2 + 9} = \frac {3}{s^2 + 3^2} $
