I think there were so many factors that all just came together brilliantly.
1. Whilst having a lot of serious aspects, it was almost this bright bubbly feel after the angst of season 2.
2. A lot of the smaller characters such as Oz, Snyder, Joyce start to come into their own more and flesh out the series even more
3. You still get the fun of high school but the characters have grown up a little in their own individual way
4. Introduction of Faith was great. Having 2 slayers in the long term (rather than just for a couple of eps like with Kendra) was a fab addition, but it was a little extra something when they took Faith to a darker side and opposite to Buffy. Especially the idea of an agent for good ending up working for the current Big Bad.
6. Which brings me to the brilliance of the Mayor. The way he flips from appearngly jovial to this cool, threatening sinister thing. Brilliant!
7. There were also so many good stand alones in this series alongside the story arcs.
That post ended up longer than I intended, but yes, I think it's a mix of even more factors than I've listed here that makes season three so popular.
Hmmm...might go watch Band Candy.