It could be easier to understand if instead using z-Transform they shown that you could do better numerical second order integration if you take few previous points into account instead of just one as in naive method of integrating twice with forward Euler integration.

Of course purpose of this lecture was probably not to make you understand what you are actually doing while numerically integrating, just to teach you how to put this stuff into simulink that has z-Transform as one of it's primitive building blocks.