I remember actually seeing that episode, and am disheartened to hear that the 20-second clip is all that survives.
If my erratic memory is to be trusted, they had some extra time and decided to just have some fun. The "solo" round occurred at the very end of the episode (again, if memory serves). And, of course, that's Lee Meriwether cheering Shatner on from the sidelines.
I should add, Shatner was given only 30 seconds to "play," but he dawdled, which is why Dick Clark comically shouts at him, "Come on, ya dummy, get over here!"