"You mostly hear the net control station. Net control is usually outside the affected area and runs an amplifier and a very good antenna setup. Stations reporting might be running low power, temporary antenna, and maybe even battery power. So you may or may not hear them. But what's important is that net control hears them, or another station is able to hear and relay to net control. This is not broadcast radio, and should not be expected to sound like it.
Also, a lot of the action is on other frequencies so as not to tie up this, the main contact frequency. Stations make contact via the net control, then move off frequency to communicate. For example, in the video you can hear net control take a checkin from a station 25 miles south of Dallas. Then you hear the National Hurricane Center (which I think is in Miami, FL) ask the net control if the station checking in has VHF or UHF contact into the affected area. VHF and UHF are used for local communications, and the Hurricane Center is well outside VHF/UHF range. The Dallas station replies that he has VHF contact only (It's a bit too far for UHF). Hurricane Center wants to talk to him further, so Net Control gives them time to make contact after which they move to a different frequency to continue their communications. While they are making contact, you can't hear the Dallas station but you can hear Net Control and you can also hear the Hurricane Center, but not as strongly."
It really is fascinating to listen to this going on, but you have to understand what is actually happening because as the viewer pointed out, you cannot hear everything. Also the video only captures a small snapshot of the big picture. In fact while I was listening, the net control announced that the hurricane was currently making landfall. Unfortunately I didn't have the camera rolling at that moment, so I started filming again and commented on that fact. This was approximately ten minutes before 9 PM Central DST.
It's worth pointing out that listeners in different areas might hear different parts of the net. A listener in South Carolina for example might hear the Hurricane Center very strongly, but hear the Net Control only weakly or not at all. The same listener might also hear the station in Dallas quite well. A listener in Houston with only a 2 Meter VHF mobile or handheld radio might hear or even be in contact with the station in Dallas and other local hams, but obviously will not hear any of the HF action.
The great thing is that any licensed ham who is interested can get involved in this sort of public service activity.