You mean besides the fact that there are seven sequels to it (five, starting with "Crossover," in Deep Space Nine and concluding with the two-part "In a Mirror, Darkly" from Enterprise), what makes this one a classic? Evil Twin Spock's goatee became a whole standard for evil twins (in Futurama Flexo, for instance). Is that good enough?
Fine. Anyway, bottom line is "Mirror, Mirror" is one of those episodes that easily transcends anything else the series was doing at the time or ever did at all, even. It was the second of a long series of such episodes from the second season, although it's among a small handful to be immediately identifiable (arguably "The Trouble with Tribbles" eleven episodes later is the leading contender for most iconic in this stretch).
It's not like the series hadn't done evil doppelgangers before, and would do so again, but never near to this effect. This is also a case of embracing the concept of alternate realities. Kirk and a few others (he's the only one that really matters in the episode, even though McCoy is in this group, surprisingly) have a transporter accident (another Star Trek trope this episode typifies) that lands them in what's since been deemed the Mirror Universe (after the episode's title), where the Federation doesn't exist. In its place is the Terran Empire. Both employ the services of Starfleet. By necessity, everything else has developed the same except how people behave (or gain rank).
Oh! And here is a good chance to remind you that Chekov has been a part of the show since the season premiere! This is bad news for him in "Mirror, Mirror," however, because he gets stuck in the agony booth (which is exactly as it sounds). Not such a (good) Russian epic this time around, Pavel.
Sulu and Uhura have bigger roles than usual, with Sulu coming out more favorably playing the most villainous person in the Mirror Universe (the only case of even a hint of the "evil Japanese" concept in the character's history, twenty years after WWII, thankfully). It's odd that the only real sign of his career ambitions before
The episode features one of the more effective lectures from Kirk about his moral superiority (it's easier than usual to agree with him, although anyone probably usually does unless they consider how condescending and anti-Prime Directive his typical attitude really is) as he explains to Mirror Spock how one man actually can make a difference.
Plus, seven sequels! That never happens! Except in this incredibly rare exception. Easily reason enough to call it a classic. And it remains very enjoyable to watch, too.
franchise * series * essential * character