This is a very good question, and it is one that I think i may not have explained in lecture or in the book.
It is true that the strongest interaction of the neutron with the solid is usually the nuclear interaction.
However, since the neutron has a dipole moment, it will interact with ANY magnetic field. The point is, that in an antiferromagnet there are local fields B(r) which are periodic and are now a function of the position in the (now larger) unit cell. So in addition to having short range interactions with the nucleus, the neutrons have a interaction with any local magnetic field. Since the magnetic field is not limited to the position of the nucleus, it is now similar to x-ray scattering in that there is some "form factor" for scattering which tells you about the distribution of scattering strength throughout the unit cell.
It turns out that while the nuclear scattering strength is usually the strongest effect, the magnetic scattering of neutrons in antiferromagnets is not that much smaller.