We locally probe the magnetic fields generated by a spin-torque oscillator (STO) in a microbar of ferrimagnetic insulator yttrium-iron-garnet using the spin of a single nitrogen-vacancy (NV) center in diamond. The combined spectral resolution and sensitivity of the NV sensor allows us to resolve multiple spin-wave modes and characterize their damping. When damping is decreased sufficiently via spin injection, the modes auto-oscillate, as indicated by a strongly reduced linewidth, a diverging magnetic power spectral density, and synchronization of the STO frequency to an external microwave source. These results open the way for quantitative, nanoscale mapping of the microwave signals generated by STOs, as well as harnessing STOs as local probes of mesoscopic spin systems.