The introductory remarks are helpful, but the real point starts at 1:07. Sagan was wise enough to frame this as an anti-Republican argument, so it got on major network television even in the Clinton era. It is actually a very serious threat to a self-governing society.
1) Science and technology are increasingly a focal point of our society, but they are not generally understood by the public -- yet in a democracy, the public has to make decisions.
2) Since they cannot, actual power passes out of their hands into the hands of 'experts' appointed by the government -- yet the same problem applies to choosing who the experts are. The people cannot assess whether real experts, or mere partisan power-players, have been chosen; and the politicians, not being experts either, can't distinguish the genuine ones either.
3) Since the politicians have to choose, and can't distinguish between real experts and political allies who are claiming to be experts, they'll generally choose political allies -- there's something in it for them there, at least. Appoint some nobody just because he has a degree or something and that person might do anything once in power. At least the party functionary will do what you want.
4) Thus, the 'scientific and technological society' ends up not only destroying self-government in favor of government by experts, but actually fails to achieve government by experts in favor of government by factional loyalists regardless of their mental or technological capacities.
Then the second reason, which relates directly to Tex's post below:
5) Science is really about skeptical empiricism, a mode of inquiry rather than a body of knowledge.
6) A free society needs people who have both education and skepticism, and must be free to question ideas. This is the only way to have a self-governing people.
7) Failing that the people do not run the government, the government runs them.
We seem to have achieved both of these in the last few years. We are now "governed by experts" to a greater degree than ever before, yet these experts at places like the Department of Energy seem to have been chosen for ideological reasons primarily (and possibly solely). All of this power has passed to the Secretary of Health and the rest of the bureaucracy, and the people in charge of it look to ordinary eyes to be barking mad, interested in political domination rather than actual expertise.
Meanwhile, attempts to question the science put forward by these experts is increasingly forbidden. Indeed, it looks like the government is investing heavily in technologies and partnerships expressly intended to suppress free inquiry. This is being done in the name of protecting less-qualified ordinary people from drawing wrong conclusions by being exposed to 'bad' information; but what it's actually suppressing is the very spirit of science Sagan described. It is the spirit of skeptical empiricism that is being attacked, to make sure there aren't challenges to the appointed 'experts' who increasingly serve as the replacement for democratic self-government.