Just to leave some space here I link to an introduction of linear transformation of state. How is it useful? Can nayone provide an example where it is indeed useful and has to be done? http://www.staff.ul.ie/burkem/Teaching/st-tr.pdf
The examples in which it is useful are already given in your PDF file.
The first reason is modal decomposition (use $x=Vz$, in which $V$ is the matrix containing the eigenvectors of the matrix $A$). Assuming we have n distinct eigenvalues, then modal decomposition helps us to turn a difficult system into a diagonal system, in which we can solve each line of the ODE by a simple integration.
The second application is the controllable canonical form. This system also has a very simple structure. Additionally, it is used to simplify the pole placement method when doing it by hand, without using the Ackermann formula.
A third application is the Kalman decomposition of a state space system into controllable & observable, controllable & not observable, not controllable & observable and not controllable & not observable subsystems.
In general, you can conclude that a linear transformation might be used to transform the system into a form which is easier to handle hand the original system or it gives you additional information that you couldn't simply see without the transformation.