There's a long tradition of ambitious people travelling to the big center of whatever they're interested in. And while NYC is more of a startup hub than it used to be, the Valley is still the center. Founders who come here find it an eye-opening experience. So we wouldn't necessarily be doing NYC-based founders a favor by establishing a branch there.
Doing YC in no way means you have to leave New York for good.
Specific: YC gave us an unbelievable network of founders, hackers and investors in the Valley, and that stayed with us even after we moved back to New York. There's absolutely no way we'd have as strong a west coast (or even east coast) network if we hadn't done YC.
Vague: Valley culture. YC is in many ways at the center of the Valley and embodies the best it has to offer. I don't know how to explain it, but there's definitely a different culture at YC and in the Valley in general than in New York or anywhere else I've been.
It's hard to quantify or describe concisely, because it's really the sum total of people's attitudes, and the conversations you overhear on the street, and the density of people who have built or are building startups, and a thousand other things.
The only analogy I can think of is traveling to another country. For instance, when I traveled to Europe, it felt extraordinarily different from the US, and the differences weren't always easy to pintpoint.
Moving out there was an incredible experience, and being able to bring what we learned there, and the network we built, back to the east coast is a huge asset for tutorspree. If you're willing to do anything to make your company succeed, then moving to the west coast for a few months weighed against everything it gives you is kindof an easy decision.
edit: PG's reasons for moving to California permanently (2.5 years ago) are described here: http://ycombinator.com/ycca.html - but as YC has grown, I'd imagine things are a bit different as there are many more people involved now.