The idea is to look at each element individually, rather than each subset. For each $k \in\{1,2, \ldots, 999\}$, there are 1000 choices for the first subset in the chain that contains $k$. This count includes the possibility that $k$ doesn't appear in any of the subsets. If $S_{i}$ is the first subset containing $k$, for some $i \in\{1,2, \ldots, 999\}$, then $k$ is also in $S_{j}$, for all $i<j \leq 999$. As a result, picking the first subset that contains $k$ uniquely determines the appearance of $k$ in all the subsets. It follows that there are $1000^{999}$ such subset chains.