(Everything is derive, all $\mathcal{D}$-modules are holonomic and $f: X \to Y$ is proper for simplicity).
We may use the adjunction $\int_f \dashv f^!$ to get some interpretation
Let $\mathcal{M}$ be a $\mathcal{D}_X$-module and $\mathcal{S}$ a $\mathcal{D}_Y$-module. Then the solution complex of $\int_f \mathcal{M}$ in $\mathcal{S}$ satisfies:
$$\mathcal{Hom}_{\mathcal{D}_X}(\int_f\mathcal{M},\mathcal{S}) \cong f_*\mathcal{Hom}_{\mathcal{D}_Y}(\mathcal{M,f^!\mathcal{S}})$$
So morally at least the solutions of $\int_f\mathcal{M}$ in some function space $\mathcal{S}$ on $Y$ correspond to the integrals (in the sense of sheaf theory) of solutions to $\mathcal{M}$ in functions pulled back from $\mathcal{S}$ along $f$.
In terms of hom-sets in the derived category one perhaps obtains an even clearer picture:
$$Hom_{\mathcal{D}_X}(\int_f\mathcal{M},\mathcal{S}) \cong Hom_{\mathcal{D}_Y}(\mathcal{M},f^!\mathcal{S})$$