There are so many...
Others have already mentioned the drama of Kirk's escape from the Constellation in "The Doomsday Machine", Kirk's realization that Edith Keeler must die, the Spock/Amanda scene, and the "Risk is our business" speech. All of them are great. But for me, the one that made me a life-long Star Trek fan...
When they seek to understand and communicate with the Horta instead of blasting her to kingdom come out of rage and self-defense. We rise above our animal self-preservation instinct to be truly human (and Vulcan).
This one. The moment when the Horta is transformed in Kirk's mind from a monster killing the miners and his crew into a mother desperately defending her children from the humans who are killing them.
There are other episodes that I love and which carry messages which have resonated me all my life, but the epiphany in "The Devil In The Dark", that the humans have been the villains all along and that they need to make amends, was incredibly powerful to me.
Yes!, the transformation of KIRK's perception and attitude,... absolutely the KEY,.... Outstanding point!
In additional effect is that it allows the HORTA to function within the story as a whole character with its own motivations and purpose, rather being trapped into functioning within the story as a 'simple monster',... stuck playing only 'monster attack' scenes.
It is at this moment when the quality of story is allow to raise yet another level in sophistication, and explore other avenues of a more meaningful purpose - which in turn raises the level of enjoyment and meaning for us, as viewers.