They show large redshifts, meaning that they are moving away from us at large velocities. Yet this tremendous power has to arise in a region that is comparable in size to the Solar system, and we know this from the fact that their brightness varies on a relatively short time scales. Since no object can be larger than the distance light can travel over a time during which the object changes its brightness by, say, a factor of two - this implies that their tremendous light output arises in a relatively small volume. We believe that the best scenario is that quasars are powered by an infall of matter onto a very massive black hole, having a mass as large as a million to 100 million Suns. So, to the first order, lifetime of a quasar has to be at least one to ten million years.
Second point has to do with the number density of quasars as a function of their distance.
At the distance corresponding to the time of roughly 9 billion years ago, we already see many galaxies and quasars, and we infer that the masses of their black holes are about 1-10 million solar masses. To account for those quasars of the past, we should have a lot of rather anonymous local galaxies harboring very massive but inactive black holes in their center. So, this can account for local quasars being dim. This implies that the quasar phase is probably shorter than a few billion years. Show me a printable version of this page.