Proving that there exists a distance $d$-dominating set of size $O(n/\delta)$
Let $S$ be a maximal set of vertices in which any two vertices are at distance at least 3. (Note maximal just means that $S$ cannot be enlarged by adding new vertices.) By design, any other vertex is at distance at most 2 from some vertex in $S$, and therefore $S$ is a 2-dominating set. On the other hand, if we define the $B_1(v)$ to consist of all vertices at distance at most 1 from $v$, then the sets $B_1(v)$ for $v \in S$ are disjoint. Since the minimum degree is  $\delta$, the size of each set $B_1(v)$ is at least $\delta+1$, hence $|S| \leq n/(\delta+1)$.