  We present a detailed study on the microscopic origin of magnetism in
suspended and dielectrically embedded CrI$_3$ monolayer. To this end, we
down-fold two distinct minimal generalized Hubbard models with different
orbital basis sets from \emph{ab initio} calculations using the constrained
random phase approximation. Within mean-field approximation, we show that these
models are capable of describing the formation of localized magnetic moments in
CrI$_3$ and of reproducing electronic properties of full \emph{ab initio}
calculations. We utilize the magnetic force theorem to study microscopic
magnetic exchange channels between the different orbital manifolds. We find a
multi-orbital super-exchange mechanism as the origin of magnetism in CrI$_3$
resulting from a detailed interplay between effective ferro- and
anti-ferromagnetic Cr-Cr $d$ coupling channels, which is decisively affected by
the ligand (I) $p$ orbitals. We show how environmental screening such as
resulting from encapsulation with hexagonal boron nitride (hBN) of the CrI$_3$
monolayer affects the Coulomb interaction in the film and how this successively
controls its magnetic properties. Driven by a non-monotonic interplay between
nearest and next-nearest neighbour exchange interactions we find the magnon
dispersion and the Curie temperature to be non-trivially affected by the
environmental dielectric screening.
