How do we know that pulsars are neutron stars?

Answer: Because they rotate really fast.

First, it is necessary to clarify what is a pulsar (pulsating star) and what is a neutron star:

A pulsar is a neutron star that emits very intense electromagnetic radiation at short and periodic intervals due to its intense magnetic field that induces this emission.

A neutron star, is the name given to the remains of a supernova. In itself it is the result of the gravitational collapse of a massive supergiant star after exhausting the fuel in its core.

Neutron stars have a small size for their very high density and they rotate at a huge speed.

So, the way to know that a pulsar is a neutron star is because of its high rotating speed.

Nevertheless, it is important to note that all pulsars are neutron stars, but not all neutron stars are pulsars.