By combining two independent approaches, inelastic neutron-scattering measurements and density-functional-theory calculations, we study the spin waves in the collinear antiferromagnetic phase (AFM2) of ${\mathrm{Mn}}_{5}{\mathrm{Si}}_{3}$. We obtain its magnetic ground-state properties and electronic structure. This study allowed us to determine the dominant magnetic exchange interactions and magnetocrystalline anisotropy in the AFM2 phase of ${\mathrm{Mn}}_{5}{\mathrm{Si}}_{3}$. Moreover, the evolution of the spin excitation spectrum is investigated under the influence of an external magnetic field perpendicular to the anisotropy easy axis. The low-energy magnon modes show a different magnetic field dependence, which is a direct consequence of their different precessional nature. Finally, possible effects related to the Dzyaloshinskii-Moriya interaction are also considered.