From Office Hours with a GGT
I am not seeing how proper discontinuity implies there are finitely many translates of $B$ that have distance at most $D$ from $B$.
Proper discontinuity says that if $K$ is compact, then $|\{g \in G \mid gK \cap K\}| < \infty$.
I take it that this can be rephrased as saying there are finitely many translates of $K$ that have distance zero from $K$?
So, there should be finitely many translates of $B$ that have distance $0$ from $B$.
How is the author concluding finitely many translates with distance at most $D$?