Perhaps you understand English better than French? You may not have noticed, but upon rewatching the video, my best guess is that Chomsky was speaking English and Foucault may have been speaking French. This may have made Foucault marginally more difficult to follow...
Joking aside, without taking away from Chomsky, at least in places I found Foucault surprisingly clear:
[O]ne of the tasks that seems immediate and urgent to me,
over and above anything else, is this: that we should
indicate and show up, even where they are hidden, all the
relationships of political power which actually control the
social body and oppress or repress it.
What I want to say is this: it is the custom, at least in
European society, to consider that power is localised in
the hands of the government and that it is exercised
through a certain number of particular institutions, such
as the administration, the police, the army, and the
apparatus of the state. One knows that all these
institutions are made to elaborate and to transmit a
certain number of decisions, in the name of the nation or
of the state, to have them applied and to punish those who
don't obey. But I believe that political power also
exercises itself through the mediation of a certain number
of institutions which look as if they have nothing in
common with the political power, and as if they are
independent of it, while they are not.
One knows this in relation to the family; and one knows
that the university and in a general way, all teaching
systems, which appear simply to disseminate knowledge, are
made to maintain a certain social class in power; and to
exclude the instruments of power of another social class.
Institutions of knowledge, of foresight and care, such as
medicine, also help to support the political power. It's
also obvious, even to the point of scandal, in certain
cases related to psychiatry.
It seems to me that the real political task in a society
such as ours is to criticise the workings of institutions,
which appear to be both neutral and independent; to
criticise and attack them in such a manner that the
political violence which has always exercised itself
obscurely through them will be unmasked, so that one can
fight against them.
This critique and this fight seem essential to me for
different reasons: firstly, because political power goes
much deeper than one suspects; there are centres and
invisible, little-known points of support; its true
resistance, its true solidity is perhaps where one doesn't
expect it. Probably it's insufficient to say that behind
the governments, behind the apparatus of the State, there
is the dominant class; one must locate the point of
activity, the places and forms in which its domination is
exercised. And because this domination is not simply the
expression in political terms of economic exploitation, it
is its instrument and, to a large extent, the condition
which makes it possible; the suppression of the one is
achieved through the exhaustive discernment of the other.
Well, if one fails to recognise these points of support of
class power, one risks allowing them to continue to exist;
and to see this class power reconstitute itself even after
an apparent revolutionary process.
Foucault believes that seemingly neutral institutions such as education and medicine act to preserve the power of one social class over another. While it's clear how the police and military maintain authority, it's less clear how topics like constant surveillance and attitudes toward sexuality would tilt the playing field. Foucault's philosophy offers a framework to analyze these different forms of power based on their effect rather than their apparent mechanism.