Why are the Knicks compelling?
Matt Glassman has been discussing (here and here) why Knicks playoff runs "ignite New York City like nothing else."
My answer here starts with how great NBA playoffs are. You get high volume and lots of intensity and drama per game. Sequences of games against the same team bring out the majesty of basketball in a way that doesn't apply to baseball (which isn't matchup-based in the same way) or hockey (where there are so many more players). It's just great. I've always been a baseball fan first, and football had a grip on me the way you'd expect given my age and upbringing, but the most intense excitement I've ever felt about a home-city playoff run was the 2003-04 Pistons, and the most exciting moment of my Detroit-sports-fan life to date is the Tayshaun Prince block.
So my first answer about the Knicks is just that NBA playoffs are awesome. But now that I've thought about it for a few days, I think things are more nuanced, and I need to speedrun some speculative cultural history to say why.
I am a child of the '80s and '90s and feel most comfortable in a certain '90s mass culture. One aspect of that was a kind of collective attention to the stories of the day. It was easier to focus collective attention on something, and the character of that attention was different. It's not easy to describe it, but it has something to do with a dominant narrative mediated by things like SportsCenter instead of reactions-to-reactions, especially of the social-media sort. All this needs to be described at length, and I'm not going to do that here, but I trust you get the idea, at least if you lived through it.
As I see it, this declined, along with most of '90s culture, after 9-11, but it persisted in that decline for many years. When I was in college, from 2001 to 2005, sports stories still basically felt the way they did in 1998, but supplemented with a lot more browsing on espn.com.
The last gasp of this whole way of processing sports, as far as I can tell,1 was Linsanity: Jeremy Lin in 2012 was the last great '90s-style sports event. Twitter very much existed, and had plenty of Lin content, but my experience and memory of it are not at all defined by Twitter, pseudoprofessional takes, meme culture, or any such thing. It was just a great story everyone was following. The term "Linsanity" is so apt because it suggests an analogy to Lisztomania, Sinatra fandom, all that Rudolph Valentino stuff, and so on. In 2015, Lin would have been a big and great story, but Linsanity couldn't have happened.
And it could only have happened with the Knicks.2 The Knicks are special and New York is special (and so expensive that Lin had to sleep on couches). Many of us had paid little attention to the Knicks for many years but had no trouble getting excited about Lin. Which makes me think that there's more to answering Matt's question than just the fact that NBA playoffs are great: there really is something about the Knicks.
P.S.: Matt has, coincidentally or not, also been posting about monoculture. It's highly relevant to all this, but I wrote everything above before I read it.