The definition of outer measure is as follows: $m^*(E) = \inf \left\{ \sum\limits_{k=1}^{\infty} l(I_k) : A \subset \bigcup\limits_{k=1}^{\infty} I_k \right\}$
If a set is bounded then for some $p\in E$ every $e \in E$, $d(e, p) \leq M$.
In my mind I interpret this as saying that every point has a finite distance less than or equal to M. However, I'm not sure how to translate that to the definition of outer measure. My first idea was that the intervals $I_k$ are the lengths between a fixed $p$ and $e$. Since they would all clearly be less than $M$, then they are all finite. Thus the outer measure must be finite.