Firstly, its not what they were filming with, but when. Get the timeing right, the clarity at 5am is totally different to the haze at 2pm. Also, you said helicopter - get a few hundred feet above the ground, again a lot of the haze will go away. Also, early in the morning, less people to make the shot look messy! Look for long shadows, that's a give away...
Once you've planned when to do the shot, checked the weather forcasts, rescheduled 3 times, and done it once - you do it again, and again, and again, until its right. Then if the budget allows, you come back with the helicopter again another day!
As for the technicalities, yes they will be using a very expensive camera and lenses - didn't see it but if it was a specific show for aerial shots, probably some form of film-quality camera instead of a standard ENG unit - but this just means you can downscale the image to make it crisper, and have a wider dynamic range for grading in Post. If on a special shoot the camera would probably have been gyro-stablised and external to the helicopter, in addition to any normal lens IS - but this won't have much to do with how clear it is, you can't magic away haze, you just need to avoid it!
