What we have here are Gene Kelly and Danny Kaye singing a medley of Kelly songs from his movies, and interwoven with a couple of dance numbers. It's not from a special -- though by today's standards, it would seem to be. Rather it's just a long, impressive and fun segment for Danny Kaye's weekly variety series. Even back in 1963, when this episode is from, it wasn't standard fare to have such a long number. But the ease and charm of the two performers just flows along.
The video is sort of odd, technically. The image quality is good, but it has a bit of a herky-jerky quality to it. Perhaps it will play better on your end, but since it's always been like this whenever I've watched it, I suspect the issue is built into the video itself. It's very watchable, though. Just not smooth.