(All functors and categories are derived, 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:
$$Hom_{\mathcal{D}_X}(\int_f\mathcal{M},\mathcal{S}) \cong Hom_{\mathcal{D}_Y}(\mathcal{M},f^!\mathcal{S})$$
(Very-)simple illustrating example: Suppose $Y = pt := Spec \mathbb{C}$, $\mathcal{M} = \mathcal{O}_X$, $\mathcal{S}=\mathcal{O}_Y=\mathbb{C}$. In that case, since $X \to pt$ is flat we have $f^!\mathcal{S} = \mathcal{O}_X$. Moreover, in general $\mathcal{End}_{\mathcal{D}_X}(\mathcal{O}_X) \cong \mathbb{C}_X$ and therefore:
$$\mathcal{Hom}_{\mathcal{D}_X}(\int_f\mathcal{O}_X,\mathbb{C}) \cong f_*\mathcal{Hom}_{\mathcal{D}_Y}(\mathcal{O}_X,f^!\mathbb{C}) \cong f_*\mathcal{End}_{\mathcal{D}_X}(\mathcal{O}_X) \cong f_*\mathbb{C}_X \cong H^{\bullet}_{dR}(X)$$
In other words, the solution complex of the de-Rham $\mathcal{D}$-module (on a point) associated with the structure sheaf $\mathcal{O}_X$ is the de-Rham cohomology of $X$.
In general when $Y$ is not a point and $\mathcal{M}$ arbitrary you will still have a Spencer-like projective resolution of $\mathcal{M}$ of the form $(\mathcal{M}\otimes \bigwedge^{\bullet} \Theta_X, \partial_M)$ and then $\mathcal{Hom}_{\mathcal{D}_X} (\mathcal{M},f^! \mathcal{S})$ will resemble a de-Rham complex. Then applying $f_*$ is a sheaf theoretic operation which corresponds to simply taking global sections of the resulting complex.