Who knows, kinda a misleading title. I don't think this was even any sort of "rescue" ship, they were probably just moored there in port. Not really their fault, though was a bit in bad taste to cheer as it went down... It looked like they crew had scuttled a little bit before he started recording anyway, that's why you see another boat right next to it start to go away.
I think they were cheering/laughing at the hypocrisy of the situation, a mighty USN ship sits and does nothing as a tug sinks right next to it. Then the OOD comes on the 1MC and says theirs nothing to be done. I doubt anyone was still on board as the thing was going under. But thanks for finding a video on YouTube from 1990 to share with us, how enlightening this was.