The very short answer is that properly applied magnetic fields, via the Zeeman effect, can cause atomic absorption bands to split into multiple bands, and these bands have different polarization properties. This is the simulation model I use to demonstrate the spliting and polarization behavior:
This model demonstrates the transverse and longitudinal Zeeman effects. The theory of Zeeman splitting is covered elsewhere, but the basic idea is this: in a magnetic field, atomic transitions may be “split”, because the levels associated with the atomic transitions are no longer degenerate (due to those pesky magnetic quantum numbers and the associated selection rules for allowed changes in them). Hence, the transition splits into several transitions. The simplest case, that of ordinary Zeeman splitting, involves a splitting into three bands: 1 $\pi$ band and 2 $\sigma$ bands. In the transverse field case, where the magnetic field is transverse to the light propagation direction, the $\pi$ band is linearly polarized parallel to the magnetic field direction. It does not spectrally shift as a consequence of the application of the magnetic field. The two $\sigma$ bands are spectrally shifted, by an amount proportional to the strength of the applied magnetic field, to each side of the $\pi$ band. The $\sigma$ bands are linearly polarized orthogonally to both the light propagation direction and to the magnetic field direction. The light propagation is the “z” direction. In the transverse magnetic field orientation, the magnetic field axis is along the “x” axis. Therefore, the $\pi$ component absorbs light that is plane polarized along the x axis. The two $\sigma$ components absorb light that is plane polarized along the “y” axis. The next figure shows the unshifted $\pi$ band in the center and the 2 $\sigma$ bands shifted out of harms way:
So the light source is linearly polarized and the $\pi$ band cannot absorb it: it could only absorb the orthogonal polarization, which is not present. The 2 $\sigma$ bands would be able to absorb the light source's polarized light, except that the magnetic field has spectrally shifted them out of harm's way. Thus, when the magnetic field is on, only the broad background absorbance absorbs light. When the magnetic field is off, the background still absorbs light and the atoms do as well.
In the anomalous splitting situation, it is possible to have rather complex splittings, even neglecting hyperfine splitting complications, isotope effects, etc. In the simplest anomalous splitting, there are two $\sigma$ bands and two $\pi$ bands. The $\pi$ bands are spectrally shifted, but are still polarized parallel with the magnetic field. Their spectral shift is less than that of the $\sigma$ components. In the longitudinal Zeeman configuration, the magnetic field is along the light propagation axis, so there is no $\pi$ component to the absorbance because light cannot be polarized along its propagation direction. The two $\sigma$ components are circularly polarized, with opposite handednesses, and are spectrally shifted by an amount proportional to the strength of the applied magnetic field. The next figure shows the splitting for calcium, in the longitudinal Zeeman configuration:
Both of these Zeeman configurations are used for background correction in atomic absorption spectroscopy. The basic idea behind all background correction schemes, in atomic absorption spectroscopy, is this: perform two absorbance measurements and take their difference. Both measurements should be designed to do an excellent job of measuring the interfering background absorbance, but only one of the measurements should do an excellent job of measuring the desired atomic absorbance. The other measurement should do a poor job of measuring the desired atomic absorbance. Differencing then discriminates against the interfering background absorbance, because it is common mode, and it yields the desired atomic absorbance, because it is differential mode. Just how well the scheme works depends on a variety of factors, so, as it happens, there are several viable background correction schemes in use.
The Zeeman effects are useful for background correction because, by suitable use of polarizers and magnetic fields, the desired atomic absorbance can be effectively “turned off”. This makes it possible to perform a measurement where the absorbance measured is almost exclusively due to just the interfering background absorbance. As noted above, this measurement is one of the two needed for the differencing involved in background correction. Hope this answers your question.