YouTube hasn't explained what prompted the surge, although it's easy to point to a few factors. For a start, the incendiary nature of the debate helped -- people around the planet wanted to catch more of those outrageous statements, especially knowing what happened in the first debate. The 9PM Eastern timing also likely drove some viewers to YouTube recaps instead of watching live TV. The rise of cord-cutting may have played a part as well, although that would be difficult to quantify.
It's important to add that this is only YouTube's data, for that matter. Twitter says that the second debate saw over 17 million tweets, the most it had ever seen for a debate -- and you know that some of those users watched the showdown on Twitter itself or other internet services, such as Facebook and news websites. TV is still a force to be reckoned with in debate coverage, and might still have an overall edge in contiguous viewing (the average YouTube viewer tuned in for nearly 25 minutes, or less than a third of the debate). Still, it's doubtful that there's any going back.
Update: We should add (thanks to reader Hussain) that internet and TV views are typically counted differently, so this isn't quite a perfect one-for-one comparison on top of factors like viewing time and on-demand programming. YouTube counts a view if you've been watching for as little as 30 seconds, while Nielsen is focused on the average viewership strung out across the length of a program. One person can register multiple views, as well. Not that TV ratings are flawless (they don't count public viewing, and sometimes rely on imperfect voluntary data), but there's a distinct chance that TV viewers were more engaged than their YouTube counterparts. The numbers are still very significant, mind you -- it shows that internet viewing is now a major component of election coverage.