Recently, experimental studies of the spin excitation spectrum revealed, like in the cuprates, a strong temperature dependence in the normal state and a resonance feature in the superconducting state. Based on these findings, we develop a model of electrons interacting with a temperature dependent magnetic excitation spectrum and apply it to analyse angle resolved photoemission and tunnelling spectra in Ba_1-xK_xFe₂As₂. We reproduce in quantitative agreement with experiment a renormalisation of the quasiparticle dispersion both in the normal and the superconducting state, and the dependence of the quasiparticle linewidth on binding energy. We estimate the strength of the coupling between electronic and spin excitations. Our findings support a dominantly magnetic pairing mechanism.