Note that $x$ is contained by $[a,b]$ if $a\le x$ and $x\le b$. That is, the point $(a,b)$ is contained in the two-sided range of all points above and to the left of the point $(x,x)$. Thus, we can use a priority search tree to enumerate the interval (viewed as two-dimensional points) in time $O(\log n+k)$, by viewing the two-sided range as a simple case of a 3-sided range.