I know David Gerrold and Harlan Ellison have an agreement: Ellison wrote the "best" episode while Gerrold wrote the "most popular" episode.
"The Trouble with Tribbles" is probably the most popular episode with people who don't identify themselves as Star Trek fans. Since there are more of "them" than there are of "us", then by those standards it's the most popular TOS episode.
It's very accessible by the casual viewer, you don't need to know much about "Star Trek" to enjoy it. It's not my favorite by any means, but it's still enjoyable to me.
I think this covers it pretty well. An additional comment; I find William Campbell woefully inadequate as a Klingon. I like William Campbell a lot, and he was perfect as Trelane, (as another poster has already pointed out above), but his performance detracted from this story.