Probability generating functions for a linear change of variable? I am struggling to understand, why, in textbook exercise on probability generating functions it says that if $X$ is a random variable (rv) with probability generating function (pgf)  $G_{X}(s)$, then for rv $Z = X + k$ for positive integer $k$ the pgf is given by $$G_{Z}(s) = s^{k}G_{X}(s).$$
Surely if we assume that the rv $X$ takes integer values $0,1,2,,,$ with probability $p_{0}, p_{1}, p_{2},...$then  the rv $Z$ takes integer values $k, k+1, k+2,...$ with corresponding probability $p_{0}, p_{1}, p_{2},...$, and thus the corresponding generating functions should be identical? The only thing that changes is the integer starting position (it is shifted by $k$) for the distribution of rv?

Since $\mathbb P(X\geqslant0)=1$ we have
\begin{align}
G_Z(s) &= \sum_{n=0}^\infty \mathbb P(Z=n)s^n\\
&=\sum_{n=0}^\infty \mathbb P(X+k=n)s^n\\
&=\sum_{n=0}^\infty \mathbb P(X=n-k)s^n\\
\end{align}
and with the change of variables $m=n-k$, this is equal to
$$
\sum_{m=0}^\infty \mathbb P(X=m)s^{m+k} = s^k\sum_{m=0}^\infty \mathbb P(X=m)s^m = s^kG_X(s).
$$
Your intuition was correct. The factor of $s^k$ is precisely what accounts for the shift of support from $\{0,1,\ldots\}$ to $\{k,k+1,\ldots\}$. Note that for example, the coefficient of $s^k$ is $\mathbb P(X=0)$, when in the probability generating function for $X$ the coefficient of $s^k$ was $\mathbb P(X=k)$.
