Take a look at this book's Chapter-$4$, Page-$159$.
It says,
Assume $0<a_0<1$. If $a_0 = 0$, then the population only grows and $0$ is not in the state space. If $a_0 = 1$, then $Z_n = 0$, for all $n ≥ 1$. We also assume that there is positive probability that an individual gives birth to more than one offspring, that is, $a_0 + a_1 < 1$.
I haven't understood this.
How come population grow if probability = $0$, and $Z_n=0$ if probability=$1$?
You have to understand what this is the probability of. The definition of $a_0$ is the probability that an individual has zero offspring.
If $a_0=0$, this means it is impossible for an individual to have zero offspring. They always have one or more offspring. So the population can never decrease in size, only stay the same or increase.
If $a_0=1$, this means individuals always have zero offspring. So the very first individual will die and produce no offspring, and thus you will have $Z_n=0$ for all $n \geq 1$.