Lawyer here. The administrative search exception to the fourth amendment means that an administrative agency can search to enforce an administrative regulation without a warrant. The most common example of this that everyone would understand is a county health department inspecting a restaurant licensed to sell food to the public.
In other words, because a restaurant has submitted to the licensing scheme of a county, it impliedly gives the county the right to search it without a warrant. Consider how a county health department could do its inspections if it had to get a warrant to do so each time it needed to.
So, in essence, in the case above, the Supreme Court is saying that since you're submitting to the government's regulations by choosing to fly, they don't need a warrant to search you.
In other words, because a restaurant has submitted to the licensing scheme of a county, it impliedly gives the county the right to search it without a warrant. Consider how a county health department could do its inspections if it had to get a warrant to do so each time it needed to.
So, in essence, in the case above, the Supreme Court is saying that since you're submitting to the government's regulations by choosing to fly, they don't need a warrant to search you.