1/(z-1) is equal to the infinite sum (z^-1) + (z^-2) + (z^-3) + ...; applied to a signal, this means the sum of all input samples of the past. This is how the digital PID controller approximates the integral over the input signal.
An in-depth explanation of the z-transform would probably be beyond the scope of this forum, and there is plenty of literature about the topic. Basically, the z-transform is as fundamental to signal processing and control in the discrete domain (where signals are time-discrete) as the Fourier- and LaPlace-transformations are for the continuous domain.
