Computationally efficient and scalable models that describe droop-controlled inverter dynamics are key to modeling, analysis, and control in islanded microgrids. Typical models developed from first principles in this domain describe detailed dynamics of the power electronics inverters, as well as the network interactions. Consequently, these models are very involved; they offer limited analytical insights and are computationally expensive when applied to investigate the dynamics of large microgrids with many inverters. This calls for the development of reduced-order models that capture the relevant dynamics of higher order models with a lower dimensional state space while not compromising modeling fidelity. To this end, this paper proposes model-reduction methods based on singular perturbation and Kron reduction to reduce large-signal dynamic models of inverter-based islanded microgrids in temporal and spatial aspects, respectively. The reduced-order models are tested in a modified IEEE 37-bus system and verified to accurately describe the original dynamics with lower computational burden. In addition, we demonstrate that Kron reduction isolates the mutual inverter interactions and the equivalent loads that the inverters have to support in the microgrid-this aspect is leveraged in the systematic selection of droop coefficients to minimize power losses and voltage deviations.