Consider one laser propagating along the +z direction. By conservation of momentum, when an atom absorbs a photon of that laser, its momentum along +z increases. After emission, the atom's momentum changes, but because emission is isotropic, the average effect will correspond to a force directed along +z.
The absorption-emission of photons I described is called photon scattering. The rate at which this scattering occurs depends on the detuning of the laser: the closer it is to resonance, the higher the scattering rate. So an atom traveling in the -z direction of that red-detuned laser will see the light as closer to resonance because of the Doppler effect. The force on the atom will be bigger.
Now take two lasers with the same red detuning, one propagation along +z and the other along -z. An atom at rest will scatter photons at the same rate from both lasers. But if it is traveling in the -z direction, it will scatter more photons from the +z laser than from the -z, so it will feel a pressure towards +z, that is, a stopping force. The converse happens if it is traveling in the other direction. Therefore, both lasers act to stop the atom, because of the Doppler effect.