In this paper we generalize the numerically exact expansion scheme for many two-level systems coupled to a bosonic (cavity) mode under open system conditions to the case of multi-level systems. For the two-level systems the method is derived from physical and intuitive arguments and the efficient numerical modeling up to hundreds of two-level systems is described. We thereby perform an analysis that allows a natural generalization to multi-level systems. The focus of the paper lies on a comprehensible view on the underlying physics. This facilitates direct application of the method and the use of additional generalizations and approximations.