Here $\overline I$ means the closure of $I$ - in general this is the smallest closed set which contains $I$. If $I \subseteq \mathbb{R}$ is an interval, then it is just the interval with the endpoints included.
For a more general example, in $\mathbb{R}^2$ you have the set $B = \{ (x,y) \in \mathbb{R}^2 : | (x,y) | < 1 \}$, the open ball of radius $1$, centred at the origin. If we take its closure we get $\overline B =\{(x,y) \in \mathbb{R}^2 : |(x,y)| \le 1 \}$, the open disc which contains the boundary. This is the two-dimensional analogue of an interval.