Photons are a penetrating probe of the hot medium formed in heavy-ion collisions, but they are emitted from all collision stages. At photon energies below 2–3 GeV, the measured photon spectra are approximately exponential and can be characterized by their inverse logarithmic slope, often called the “effective temperature” Teff. Modeling the evolution of the radiating medium hydrodynamically, we analyze the factors controlling the value of the Teff and how it is related to the evolving true temperature T of the fireball. We find that at the energies available at the BNL Relativistic Heavy Ion Collider and the CERN Large Hadron Collider most photons are emitted from fireball regions with T∼Tc near the quark-hadron phase transition, but that their effective temperature is significantly enhanced by strong radial flow. Although a very hot, high-pressure early collision stage is required for generating this radial flow, we demonstrate that the experimentally measured large effective photon temperatures Teff>Tc, taken alone, do not prove that any electromagnetic radiation was actually emitted from regions with true temperatures well above Tc. We explore tools that can help to provide additional evidence for the relative weight of photon emission from the early quark-gluon and late hadronic phases. We find that the recently measured centrality dependence of the total thermal photon yield requires a larger contribution from late emission than presently encoded in our hydrodynamic model.