The energetic and entropic interactions governing the attraction between like-charged colloidal particles grafted with oppositely charged polyelectrolyte chains in a monovalent electrolyte are investigated computationally. We employ coarse-grained models of the colloids with varying surface and polyelectrolyte charges and Monte Carlo simulations to compute the potential of mean force between two colloidal particles as a function of their separation distance. By categorizing the potentials as attractive or purely repulsive, we obtain the extent and location of the attractive-force regime within the two-dimensional parameter space comprised of the colloid surface and polyelectrolyte charge. The attractive regime is found to occupy the inside of a hyperbola in this charge space, whose shape and size is determined by a complex interplay between energetic and entropic interactions. In particular, we find that the strength of attraction at short distances is governed by a balance between favorable energetic and entropic terms arising from polymer-bridging interactions, unfavorable energies arising from the mutual repulsion or the colloid surfaces and polyelectrolyte chains, and unfavorable entropies arising from the overlap and crowding effects of chains confined between the colloid surfaces. A phenomenological model is proposed to explain the hyperbolic shape of the attractive regime and make useful predictions about changes in its shape and location for conditions not investigated in this study.