One way to think of twisted $D$-modules that I like is to view them as monodromic $D$-modules (see Beilinson, Bernstein A Proof of Jantzen Conjectures section 2.5, available as number 49 on Bernstein's web page ). Let $T$ be a torus, and let $\pi: \tilde{X} \to X$ be a $T$-torsor. The sheaf of algebras $\tilde{D} = (\pi_* D_{\tilde{X}})^T$ has center $U(\mathfrak{t}) = S(\mathfrak{t})$, and its category of modules is the category of weakly $T$-equivaraint $D$-modules on $\tilde{X}$. For any $\chi \in \mathfrak{t}^\vee$, there is a maximal ideal $m_\chi \subset S(\mathfrak{t})$, and the algebra of $\chi$-twisted differential operators is $\tilde{D}/m_\chi \tilde{D}$.
If you want to twist by a fractional power $c$ of a line bundle $L$, then you can let $T$ be the usual one dimensional split torus, $\tilde{X}$ be the total space of $L$ with the zero section removed, and $\chi = c$. For ordinary differential operators, set $\chi = 0$. Intuitively, I think of a ($\mathcal{O}$-coherent) twisted $D$-module as a sheaf vector bundle on the total space of the torsor whose such that flat sections have to obey a twisted equivariancefixed monodromy when parallel transported in the torus direction.
One way to think of twisted $D$-modules that I like is to view them as monodromic $D$-modules (see Beilinson, Bernstein A Proof of Jantzen Conjectures section 2.5, available as number 49 on Bernstein's web page ). Let $T$ be a torus, and let $\pi: \tilde{X} \to X$ be a $T$-torsor. The sheaf of algebras $\tilde{D} = (\pi_* D_{\tilde{X}})^T$ has center $U(\mathfrak{t}) = S(\mathfrak{t})$, and its category of modules is the category of weakly $T$-equivaraint $D$-modules on $\tilde{X}$. For any $\chi \in \mathfrak{t}^\vee$, there is a maximal ideal $m_\chi \subset S(\mathfrak{t})$, and the algebra of $\chi$-twisted differential operators is $\tilde{D}/m_\chi \tilde{D}$.
If you want to twist by a fractional power $c$ of a line bundle $L$, then you can let $T$ be the usual one dimensional split torus, $\tilde{X}$ be the total space of $L$ with the zero section removed, and $\chi = c$. For ordinary differential operators, set $\chi = 0$. Intuitively, I think of a twisted $D$-module as a sheaf on the total space of the torsor whose sections have to obey a twisted equivariance.