We propose that the ultra-diffuse galaxy (UDG) population represents a set of satellite galaxies born in ∼1010-1011 M⊙ haloes, similar to field dwarfs, which suffer a dramatic reduction in surface brightness due to tidal stripping and heating. This scenario is observationally motivated by the radial alignment of UDGs in Coma as well as the significant dependence of UDG abundance on cluster mass. As a test of this formation scenario, we apply a semi-analytic model describing the change in stellar mass and half-light radius of dwarf satellites, occupying either cored or cuspy haloes, to cluster subhaloes in the Illustris-dark simulation. Key to this model is results from simulations that indicate that galaxies in cored dark matter haloes expand significantly in response to tidal stripping and heating, whereas galaxies in cuspy haloes experience limited size evolution. Our analysis indicates that a population of tidally stripped dwarf galaxies, residing in cored haloes (such as those hosting low surface brightness field dwarfs), is able to reproduce the observed sizes and stellar masses of UDGs in clusters remarkably well.