According to Wikipedia:
"If $B$ is a block then $gB$ is a block for any $g$ in $G$. If $G$ acts transitively on $X$, then the set $\{gB \mid g \in G\}$ is a block system on $X$."
i.e., $\{gB \mid g \in G\}$ partitions $X$.
I am trying to understand how you can go about proving this statement. I understand that the existence of a block system in $X$ defines partitions in $X$, but how can you deduce from that a group action also leads to partitions?