Question

why do we perform inverse z transform.what is the use of it.

Answer 1

It allows you to go from the Z-plane complex frequency-domain representation to a time-series, e.g. deriving a time-series signal from a transfer function.

Answer 2

For designing a electronic system , it is important to check its behavior in both domains time domain and frequency domain . For continuous time domain signal there is laplace transform , similar for discrete signals there is z-transform.
Frequency domain analysis is also important to check whether a system is stable or not. means if u are designing an amplifier , it may behave as a oscillator at some frequency if it not perfectly designed. So these type of transforms are used there.