In conventional superconductors, the most direct evidence of the mechanism responsible for superconductivity comes from tunnelling experiments, which provide a clear picture of the underlying electron-phonon interactions. As the coherence length in conventional superconductors is large, the tunnelling process probes several atomic layers into the bulk of the material; the observed structure in the current-voltage characteristics at the phonon energies gives, through inversion of the Eliashberg equations, the electron-phonon spectral density alpha2F(omega). The situation is different for the high-temperature copper oxide superconductors, where the coherence length (particularly for c-axis tunnelling) can be very short. Because of this, methods such as optical spectroscopy and neutron scattering provide a better route for investigating the underlying mechanism, as they probe bulk properties. Accurate reflection measurements at infrared wavelengths and precise polarized neutron-scattering data are now available for a variety of the copper oxides, and here we show that the conducting carriers (probed by infrared spectroscopy) are strongly coupled to a resonance structure in the spectrum of spin fluctuations (measured by neutron scattering). The coupling strength inferred from those results is sufficient to account for the high transition temperatures of the copper oxides, highlighting a prominent role for spin fluctuations in driving superconductivity in these materials.