It's hard to say whether or not this is a glitch. I think it's rather weird that the player falls off the screen at the end, which makes me believe that it could be a glitch. But I don't know if I would just "forget" about the possibility that the rat could follow you down if I were the programmer. That makes me a bit skeptical.
My thinking was the end of the level was triggered by finishing off the Boss and they didn't expect you to actually kill the rats but instead let them fall off the screen. And you could end the level early because it wasn't triggered by the boss dieing, just any enemy. Just a thought.
This is definitely a glitch, Battletoads have plenty of them. It looks hard to pull off tough, almost as hard as finishing the level (which I have done 1 time, and am very proud of).
Excuse-me but I really don't see what is weird with that video.
I've never played at Battletoads so the video didn't appear to be exceptional for me...
The part you are missing is normally you have to race something like 3 rats. If the rat beats you down to the bomb you lose. However it appears this guy figured out how if you do what he does in the video to beat the rat until it dies, rather than falling off the screen, the level ends before you reach the boss at the bottom. Basically it makes a tough level alot shorter.