How is three-dimensional laser cooling possible?

In laser cooling, the laser light is red detuned, meaning it has lower energy than the atomic transition. An atom at rest cannot absorb it:

However, a moving atom now sees Doppler shifted laser beams. The light coming towards it from the right will be on resonance (for some velocity class) and will be absorbed:

The photon emitted by the laser has energy $hbar omega < hbar omega_0$. But when it is re emitted by the atom, however, it will have energy $hbar omega_0 > hbar omega$ ! So energy is conserved, but the emitted photon takes away some energy from the atomic cloud.

What about momentum?

The atom receives a momentum kick upon absorbing the photon, and another one (essentially of equal magnitude) when spontaneously re-emitting it. BUT the absorbed photons always come from the same direction (the laser beams) whereas the spontaneously emitted photon is random. Over time, the random spontaneous emission averages to zero, only giving you a decrease in the momentum along each laser beam direction.

So for laser beams in $6$ orthogonal directions ($pm x, pm y,$ and $pm z$) you get cooling in all directions.

Limit of the above

This kind of "simple" laser cooling works until the Doppler temperature, set by the natural linewidth of the atom $Gamma$: when the Doppler shifted frequency between right and left photons is less than $Gamma$, the atom does not know which one to absorb because it cannot resolve it.

Eventually, the spontaneously emitted photon and the resulting momentum kick does limit the temperature you can reach, and that is called the recoil limit. Which is why, to get colder with light, you need to use conservative potentials and hence not rely on scattering.

Applications to cold atoms

One of the main applications of laser cooling is to reach quantum degeneracy.

The degeneracy parameter $D$ goes as $exp(-S)$ where $S$ is the entropy. To get quantum ($D sim 1$), it is not enough to lose energy, you also need to lose entropy.

The incoming photon from the laser is coherent, hence has low entropy. The spontaneously emitted photon is random, hence has higher entropy. So you are also extracting entropy from the cold atomic gas.