The local monodromy around $D_i$ can be obtained by taking a $\eta$ a geometric generic point of $D_i$, $R$ the etale local ring of $X$ at $\eta$ with uniformizer $\pi$, then pulling $\mathcal L$ back to $\operatorname{Spec} R[ \pi^{-1} ]$. We then obtain a representation of the étale fundamental group of $\operatorname{Spec} R[ \pi^{-1} ]$. The image of this representation is called the local monodromy.

Rather than $U \cup D_i$, it would be more precise to say $X \setminus \cup_{j\neq i} D_j$. To check that $\overline{\mathcal L}$ restricted to this space is a local system, we first check that for $j \colon U \to X \setminus \cup_{j\neq i} D_j$ the open immersion, $j_* \mathcal L$ is a local system, and then that $j_* \mathcal L$ is the restriction of $\overline{\mathcal L}$.

The pullback of $j_*\mathcal L$ to $\operatorname{Spec} R$ is the pushforward from $\operatorname{Spec} R[ \pi^{-1} ]$ to $\operatorname{Spec} R$ of the restriction of $\mathcal L$ to $\operatorname{Spec} R[ \pi^{-1} ]$. So it is the pushforward of a constant sheaf from the generic point to the whole spectrum and thus is a constant sheaf. Thus $j_* \mathcal L$ is locally constant at the generic point of $D_i$. Thus it is locally constant on some neighborhood of the generic point. The complement of the largest open set on which $j_* \mathcal L$ is locally constant is a closed set contained in $D_i$ but not containing the generic point of $D_i$ and thus has codimension $\geq 2$.

If that complement is nonempty, consider a generic point $\eta'$ of that complement and a Henselian local ring $R'$ at $\eta$. The lisse sheaf gives a representation of the fundamental group of the punctured spectrum of $R'$, which by purity (since the codimension is $\geq 2$) is trivial. So again the sheaf is the constant sheaf and the pushforward is just the constant sheaf, contradicting the assumption that the sheaf is not locally constant at $\eta$, so the complement is indeed empty and $j_* \mathcal L$ is lisse.

Since the intermediate extension is the unique perverse extension with no irreducible components supported outside $U$, and $j_* \mathcal L$, being lisse, is perverse and has no irreducible components supported outside $U$, $j_* \mathcal L$ must give the intermediate extension.