Generally in a ferromagnet each atom has $Z-1$ paired electrons, with zero net spin, and one unpaired electron. The unpaired electron spins on neighboring atoms like to be parallel, which is why the material has a bulk magnetization.
If you try to inject another electron into the ferromagnetic material, that other electron has to go somewhere —
into an orbital around some atom. If it wants to go into the lowest-energy orbital, its spin has to be antiparallel to the spin of the unpaired electron that's already there. If the spins of the injected are fixed to be parallel to the unpaired electron spins in the material, then all of the lowest-energy orbitals appear "full," and the injection requires more energy. Effectively the ferromagnet has a different work function for different electron spins.
Complicating any handwavy discussion of this is the fact that the electron magnetic moment is negative.
I'm not a solid-state guy, so I invite others to correct me …
The review paper
you linked in the comments calls the source and drain materials "half-metallic ferromagnets." I think this means that the electrons in the conduction band all have one polarization, and electrons with the other polarization have Fermi energies in the band gap. In that case all of the conduction would have to take place via electrons with the correct polarization.
I don't know whether all ferromagnets are "half-metallic" like this, or whether those materials must be engineered. If they're engineered, I don't know whether it's possible to tune which spin state conducts. Very interesting!
Here is a page about half-metallic ferrormagnets
with figures and references. In particular the first figure shows the differing densities of states for majority- and minority-spin electrons in one material:
I think that producing these sorts of diagrams for a real material involves lots of solid-state black magic, but the fact that a gap appears at the Fermi energy for minority-spin electrons does mean that all the conduction electrons will be polarized.