There is no one simple explanation.
First, Kilgore was a mediocre candidate (I posted this before), so he neither galvanized his base nor attracted others outside his base.
I'll agree with this. From what a saw/read, Kaine out campaigned Kilgore and generally held better bearing and charisma/likability throughout the two's public appearances and debates.
Second, more importantly, in Virginia (as elsewhere across the nation) the left saw this year's election as a chance for revenge for their losing streak from 2002-2004. They turned out whereas conservatives were lackidaisical.
Fourth, Kaine sucessfully hid behind Warner. Warner was a very popular moderate Governor, whereas I believe time will tell that Kaine will end up being a very unpopular liberal Governor.
The left see every election as a "chance for revenge" the same way Republicans see the same ones as a chance to bury the left. I'm not so convinced that this was a factor at all. Kaine may also have "hid behind Warner" as you suggest, but when current status quo is generally accepted and actually desirable, you'd have to be an idiot to promise change or simply not follow in the footsteps of your predeccessor. Look at what happened to Gore 2000. Kaine had Warner on his side, but I suppose(and Virginians could answer this better than me) it would be prudent to ask if Allen had any influence on this election and why/why not.
Third, the media played a major role in supressing stories harmful to Kaine. A case in point, Carlos H. Bustamente Medieta (an illegal immigrant) brutally mudered Hak Bon Kim (a legal immigrant) in northern Virginia just a few weeks ago. This story was quickly hushed up.
The old biased media strikes again. You can't honestly believe supressing that story made a difference in this campaign or was some kind of consperiacy one way or the other. I suppose you'll tell me the Whitewater investigation findings and results or the impeachment hearings coverage in 1998 had no bearing on the election of 2000. If the media is biased as you habitually claim, they never would have covered it as they did.