I'm mostly looking for a little more context into this statement and to the political climate at the time, as the "Vidal vs Buckley" era around 1968 is sometimes portrayed as a pivot point in American political theater.
Why was Buckley's statement met with applause from the crowd? Was the establishment of only two political parties important to conservatives in the US during the mid- to late-1960s? If so, why?
I would disagree with /u/kilato99. The winner of that election was, in fact, John Lindsay, a Republican. Buckley's point was that John Lindsay, while ostensibly a Republican, was no more a conservative than his democratic opponent. That the choice in New York City was not right-left but left or left-er. Indeed, Lindsay would later become a Democrat in 1971.
You can hear Buckley's explanation of this here at about the 1:50 mark: https://www.youtube.com/watch?v=aMl2TVdzFa0
This was part of the overall project of "movement conservatism" as distinct from simply Republican party partisanship, of which Buckley played a critical part. Buckley did not attempt to run as a Republican in the New York City mayoral primary, nor would he have won had he done so. The point of his run, as with National Review and the overall movement conservative project was to mold the Republican party into a truly conservative party. They scored their first victory in the nomination of Barry Goldwater in 1964, but then immediately suffered one of the worst electoral defeats in American political history, with LBJs 486 electoral votes to Goldwater's 52. The triumph of his effort was the election of Ronald Reagan in 1980.
edit: typo.
edit 2: this article by Sam Tanenhaus, (who incidentally from what I understand came to have an odd and combative relationship with former colleagues on the right but as far as I can tell doesn't seep into into this piece) covers the campaign and Buckley's motivations quite well: http://www.nytimes.com/2005/10/02/magazine/the-buckley-effect.html