Mature neutron stars (not proto-neutron stars right after supernova explosions) are "relatively" cool if we compare their typical temperatures with the Fermi temperature of their constituents (the Fermi temperature increases with density, that's why is very high in neutron stars, see e.g. the pedagogical notes Neutron Stars for Undergraduates).
Since the Fermi temperature of neutrons is much higher than the actual temperature (exactly like for metals on Earth), an effective zero-temperature treatment is a good approximation in many cases: "the neutrons are degenerate", in the sense that their temperature is low enough that the effect of Fermi statistics is very important.
In this regime of density and temperature, degenerate neutrons can become superfluid via the Cooper pairing mechanism, which is the same mechanism responsible for the "terrestrial" fermionic superfluidity of electrons in superconductors, Helium-3 or other fermionic ultra-cold gases.
Cooper pairing of neutrons (neutron-neutron pairs) and protons (proton-proton pairs) in neutron stars is due to some attractive channels of the nuclear force (see also this and this for the nature of the strong interaction between nucleons).
For more technical details, this is a good review on superconductivity and superfluidity in nuclear systems: Superfluidity in nuclear systems and neutron stars. E.g., here you can find why there are no neutron-proton Cooper pairs and all the details on the attractive channels of the nuclear force at different baryon densities.