Deep State is simply short hand for the permanent bureaucracy, the faceless people that persist administration to administration; the shadow government that push the levers of power. You know what Frank’s talking about, the one’s who punch the clock at the sausage factory.
In large part it is who Reagan meant, with this well known quote:
"In this present crisis, government is not the solution to our problem; government is the problem."
January 20, 1981: From Reagan's Inaugural Address.