The videos are from CCTV5, not NBC.
It doesn't matter. Skate America is put on and funded by USFSA (is that right?) in any event one of the many ways in which they make money out of it is by selling the intellectual property rights required to broadcast the event.
Think of it this way - in order to view the event you would have to buy a ticket. Unless you pay the fee you don't get to watch it. Sounds fair enough?
Now if anybody not able to attend the event wishes to see it, it could be taped. Ok but they get to see it without buying a ticket. Fair enough? I guess as a tv figure skating fan one might say yes, as the event organiser who makes their money primarily from people seeing the event not so. Event organisers therefore sell exclusive (no-one else including themselves can broadcast) sole (the person being granted the rights and themselves can broadcast) or non-exclusive (the event organiser reserves the right to grant as many people rights to braodcast as it likes) broadcast rights to the event.
Looking at it in this way you can observe many things. The first is that selling exclusive rights to one person usually means a bigger fee since no-one else will be allowed to broadcast. You can also see that the exclusive licensee would become fairly angry at having paid a big fee for the rights and find that someone else is broadcasting it. They might then go back to the event organiser and seek some clarification that they are not in breach of their exclusive contract by licensing it to someone else.
It all depends on the terms of the NBC contract whether NBC will assert that their copyright is violated, or whether event organiser's copyright is violated.
In any event looking at the broadcast as above it should help to see that there are some parties with intellectual property rights over the broadcast and their rights are being violated.
And about the private spectator taping performances and then making them available online. I'd be almost certain that the event organiser or arena's policy (and the terms and conditions accepted in buying the ticket to attend) will clearly state that taping any of the performances is illegal.
Hope that helps a little to understand.
Ant