We investigate the bilayer Ruddlesden-Popper iridate Sr3Ir2O7 by temperature-dependent angle-resolved photoemission. We find a narrow-gap correlated insulator, with spectral features indicative of a polaronic ground state, strikingly similar to that observed previously for the parent compounds of the cuprate superconductors. We additionally observe similar behaviour for the single-layer cousin Sr2IrO4, indicating that strong electron-boson coupling dominates the low-energy excitations of this exotic family of materials, and providing a microscopic link between the insulating ground states of the seemingly-disparate 3d cuprates and 5d iridates.