Why do astronomers think that quasars are relatively short-lived phenomena?
Quasars are relatively short lived because they have a limited amount of fuel.
Most large galaxies have a supermassive black hole at their centre. If there is material in the vicinity of the supermassive black hole it will form an accretion disc which is material spiralling into the black hole. If there is enough material falling into the accretion disc it will get superheated by friction between particles and gravitational effects from the black hole. This superheated material emits vast amounts of energy which is a quasar.
The quasar can only be active while there is sufficient material falling into the accretion disc. Typically only young galaxies and galaxies which have collided with another have enough material to power a quasar. Hence quasars are relatively short lived.
The known quasars are young galaxies which are billions of light years away. They were formed in the early Universe and their light is now reaching us. These quasars will have stopped emitting a long time ago.