Neutron star interiors have not been directly measured. Instead, we use a combination of a less direct but no less valid means to ascertain their interior structure and composition.
The general structure of neutron stars is understood through a combination of stellar structure physics and the properties of their progenitors, the cores of massive stars. Ultimately this is a combination of our knowledge about general relativity, fluid mechanics, and nuclear physics. These models are quite sophisticated, and predict a number of interesting behaviors, including superconducting regions and superfluidity in their cores.
These models must match what we know about neutron stars observationally. One way to do so is to study the predictions of these models for neutron star cooling, which can be compared with observations of real neutron stars. Pulsars and magnetars are also thought to be neutron stars, and their properties depend on their interior structure.