Of all of my memories of watching TOS in my youth, there were two that stick by me the most.
The first was sitting down to watch it with my brother on October 23, 1983 when I was six years old. Just after it started, there was a special news bulletin about some dumb bomb exploding in some place I'd never heard of and my brother- much older than me- kept telling me to be quiet and stop complaining so he could hear the news. Right as the bulletin ended, the credits for Star Trek started playing. It made me cry.
The other one was seeing those aliens for the first time and thinking, "I guess aliens don't have to look like us." It was a profound thought for a child no more than eight years old.