This analogy may help: Imagine a piston at $+\infty$ moving leftward until it stopped by the presence of the sequence $a_1,a_2\ldots$. The place will stop is the $\sup$ of the sequence, which in your notation is $\overline s_1$, i.e., the first element of the sequence $(\overline s_n)$. Now let remove the first element $a_1$ of the original sequence, this may or may not change the $\sup$ of the sequence. If will change the $\sup$, the piston slip leftward to a new point $\overline s_2$. If not, then the piston will not move and $\overline s_2$ will be the same as $\overline s_1$. In any case $\overline s_2$ cannot be greater than $\overline s_1$. Then we remove the second element $a_2$, causing the piston to slip a little more. If we keep doing this the piston will keep slipping, but there will be a point where cannot go any further and this is what we call the limit superior of the sequence.
Now more formally the sequence $(\overline s_n)$ is non-increasing, i.e., is monotone, so the limit of the sequence always exists in the extended real numbers, and this limit, which is the infimum, is what we call the limit superior.