If $(Z_n)_{n\in\mathbb N}$ is an i.i.d. process with values in a normed $\mathbb R$-vector space $E$, then $$W_n:=\sum_{i=1}^nZ_i$$ is called random walk with step distribution $\mathcal L(Z_1)$.
Now let $\lambda$ be a finite measure on $E$, $\alpha>0$, $W_0:=0$, $(W_n)_{n\in\mathbb N}$ be a random walk with step distribution $\frac1\alpha\lambda$, $(N_t)_{t\ge0}$ be a càdlàg Poisson process with intensity $\alpha$ independent of $W$ and $$X_t:=W_{N_t}\;\;\;\text{for }t\ge0.$$
What does it mean and how can we show that "the jumps of $X$ occur at the jump times of $N$"? And how do we see that "the sizes of successive jumps are" $Z_1,Z_2,\ldots$?
Clearly, $n$th arrival time of $N$ is given by $$\tau_n:=\inf\{t\ge0:N_t=n\},$$ $N_{\tau_n}=n$ and $\tau_n-\tau_{n-1}$ is exponentially distributed with parameter $\alpha$.
Now my problems of thinking about the question starts with the definition of "the jumps of $X$". We may consider $\tau_0^B:=0$ and $$\tau_n^B:=\inf\{t>\tau_0^B:\Delta X_t\in B\}$$. Maybe the "jumps of $X$" are $\tau_n^{E\setminus\{0\}}$? If so, how do we obtain the claims?
If there is no jump of $N$ in the interval $[s,t]$ then $N_s=N_t$ and so $$ X_s=W_{N_s}=\sum_{i=1}^{N_s}Z_i=\sum_{i=1}^{N_t}Z_i=W_{N_t}=X_t\,. $$ In other words, $X$ has no jump in $[s,t]$. Conversely, if there are $n$ jumps of $N$ in $[s,t]$ then $N_s+n=N_t$ and $$ X_s=W_{N_s}=\sum_{i=1}^{N_s}Z_i\,,\quad X_t=W_{N_t}=\sum_{i=1}^{N_t}Z_i=\sum_{i=1}^{N_s+n}Z_i=\quad X_s+\underbrace{\sum_{i={N_s}+1}^{N_s+n}Z_i}_{\textstyle\text{jumps of $X$ in $[s,t]$}}\,. $$ In other words, $X$ has $n$ jump in $[s,t]$.