We report experimental observation of resonances excited by nonlinear localized states (rotobreathers) in Josephson junction ladders. The rotobreathers are found to persist in a frequency range that allows for their resonant interaction with linear electromagnetic modes in the ladders. This interaction leads to nearly constant voltage steps on the current-voltage characteristics. We also present numerical simulations that agree well with experimental data and confirm the resonant interaction between breathers and linear waves. Resonances occur at the base frequency as well as higher harmonics of the linear modes. The observed substructures on the resonances are attributed to the cavity modes for the ladders. Both experimental and simulated current-voltage characteristics show good quantitative agreement with an analytically calculated dispersion relation for linear electromagnetic modes.