Let $A_n\subset \mathbb R$ for all $n=1,2,3,\ldots$ and let $\sum\limits_{n=1}^\infty m^*(A_n)<+\infty$. If $E=\{x:x\in A_n\text{ for infinitely many }n\}$, then we want to show that $m^*(E)=0$.
From definition, $E\cap A_n\neq \emptyset$ for infinitely many $n$. But what this information is going to help?
A point $x$ is in infinitely many $A_n$ iff for all $k$ there is an $n>k$ such that $x\in A_n$, that is, iff $x\in\bigcap_k\bigcup_{n>k} A_n$. The result follows easily from this description, using that $m^*(\bigcup_{n>k}A_n)\le \sum_{n>k}m^*(A_n)\to0$ as $k\to\infty$.