Once it was realized that the pulse period tells us the rotation rate of the pulsar, why did astronomers know that these could not be normal stars?

Pulsar is also called as neutron stars or white dwarfs. These are formed after the supernova explosion. Their mass is higher than a normal star. And the most different thing we find in pulsar is that, in a pulsar the proton and electrons combine themselves to form a neutron. That's why this is also called as neutron star. After neutron or white dwarf the star goes to the black dwarf stage.

Remember when the core of the iron becomes iron rich that time it cannot sustain further nucleo synthesis process, that's why it goes for a supernova explosion. Which results the formation of neutron star or white dwarf, followed by black dwarf or black hole.