The pairing mechanism in iron-based superconductors is believed to be unconventional, i.e. not phonon-mediated. The achieved transition temperatures Tc in these superconductors are still significantly below those of some of the cuprates, with the exception of single layer FeSe films on SrTiO3 showing a Tc between 60 and 100 K, i.e. an order of magnitude larger than in bulk FeSe. This enormous increase of Tc demonstrates the potential of interface engineering for superconductivity, yet the underlying mechanism of Cooper pairing is not understood. Both conventional and unconventional mechanisms have been discussed. Here we report a direct measurement of the electron-boson coupling function in FeSe on SrTiO3 using inelastic electron scattering which shows that the excitation spectrum becomes fully gapped below Tc strongly supporting a predominantly electronic pairing mechanism. We also find evidence for strong electron-phonon coupling of low energy electrons, which is however limited to regions near structural domain boundaries.