The continued demand for smaller, faster, and more energy-efficient technologies has driven condensed-matter research towards materials combining multiple responses within a single phase. Particularly interesting examples are multiferroic materials featuring cross-couplings between order parameters. Ni3TeO6 is a prominent member of this material class, exhibiting a hysteresis-free transition under magnetic field which simultaneously affects the magnetic structure, electric polarization, and phonon frequencies. Here, we uncover the microscopic origin of this multiferroic transition using a combination of neutron scattering techniques. We establish the presence of a finite Dzyaloshinskii-Moriya exchange term giving rise to non-reciprocal, chiral spin waves which soften in magnetic field. Their condensation alters the magnetic structure which controls the electric polarization and drives an ionic displacement, thereby coupling the magnetic, electric, and lattice degrees of freedom. Our study demonstrates that neutron scattering is a powerful technique to elucidate microscopic interactions in multiferroic materials, informing the design of next-generation electronic devices.