On selection rules:
Say the Zeeman effect targets an atom with a ground state of angular momentum $J=0$ and an excited state of angular momentum $J=1$, such that the magnetic field splits the excited state into 3 sublevels corresponding to $M=0,\pm1$. The selection rules for transitions to/from the excited states are imposed by conservation of angular momentum: Transitions to the $M=0$ state do not involve a change in the angular momentum projection along the field direction and the photons absorbed/emitted are linearly polarized along the direction of the magnetic field and propagate perpendicular to the magnetic field. On the other hand, transitions to/from the states $M=\pm1$ carry a change in angular momentum projection. For an intuitive analogy, it is as if the excited atom becomes a rotating dipole and carries a circular current. Conservation of angular momentum requires then that the absorbed/emitted photons must propagate along the magnetic field direction and be circularly polarized: clockwise circular for the $M=+1$ transition, anti-clockwise circular for the $M=-1$ one. See for instance these notes.
On using the Zeeman effect in magneto-optical traps:
Imagine a 1d trap. Away from the center of the trap the non-uniform magnetic field induces a Zeeman effect proportional to the distance from the trap's center. The circularly polarized laser beams are oriented along the magnetic field, in opposite directions and are tuned slightly below the Zeeman transitions at a certain field intensity. Say the clockwise polarized beam is set toward the positive direction, while the anti-clockwise one is toward the negative. If an atom moves away from the trap center in the positive direction, it will eventually reach the point where its Zeeman splitting matches the frequency of the laser field. But at that position, to the right, the anti-clockwise laser field dominates and the atom is excited in its $M=-1$ state. When it re-emits scattered photons, the corresponding radiation pressure pushes it back into the trap. Similarly for atoms moving in the negative direction, where the clockwise laser excites the $M=+1$ transition. For further details see, for example, this lecture.