Why do neutrons have magnetic moments with spin?

A neutron is a composite particle consisting of three pointlike quarks. These carry fractional electric charge which adds up to zero, so the neutron has no net charge. But since the quarks do carry nonzero charges, if they are spinning around inside the neutron then it is possible for the neutron to possess a magnetic moment while not possessing a net charge.

The fact that the neutron has a nonzero magnetic moment despite having no net charge while also possessing a well-defined radius were clues that it had internal structure (the quarks), which was proven in the deep inelastic scattering experiments done at SLAC in the late 1960's.