This episode is sloppily plotted. The characters encounter a peculiar issue (as they usually do) and the entire episode banks upon our being interested in the extreme peculiarity of that issue.
The Enterprise discoveres a duplicate Earth (a setup left entirely unresovled) which has been deserted save for the children, who have survived for hundreds of years. The action of the episode involves the crew stumbling upon crumbs of information to finally (and arbitrarily) conclude that the inhabitants of this planet created a virus, the effects of which preserve the life of the children and brutally transform and kill the adults.
The only drama stems from the fact that the children will eventually reach puberty and succumb to the virus. Meanwhile, the crew is infected, and has only a limited window of sanity to produce an antidote. The resolution of the situation is so unearned and unsatisfying I'm left breathless. Is this Star Trek?
McCoy finds some research papers and creates an antidote -- fatal if he's wrong -- which he tests on himself. It works and saves everyone. Gee, that was rather easy. The only other complication (not to suggest that more would ensure better drama) is the lost-boys-like child congregation's disdain for the crew leading them to steal their communicators. Which, really, births McCoy's necessity to test the unconfirmed antidote on himself. So, it's really just the one thing. There's a nice but ultimately inconsequential plea from Kirk for the children's help.
In retrospect, Kirk's relationship with the puberty approaching Miri is ... uncomfortable. Also, this episode has two -- count 'em, two -- idiosyncratic or noteworthy Spock lines. "It could be a beaker full of death" and my personal favorite:
"That little girl is at least three hundred years older than you are, yeoman ... think about it."