The Enterprise responds to a distress call on an uncharted planet. A landing party beams down to an unconvincing studio set where they encounter a humanoid man named Rojan (Warren Stevens) and a very hot woman in a backless outfit named Kelinda (played by sexy giallo babe Barbara Bouchet), who demand that Kirk surrenders his ship. Naturally, the captain isn't about to do so, but when he refuses the aliens paralyse the landing party using devices worn on their belts. Rojan explains that he and Kalinda are Kelvans from the Andromeda galaxy, on a mission to find planets in the Milky Way suitable for conquest .
Held prisoners in a cave, Kirk and his crewmates attempt to escape but are quickly apprehended. As punishment, the two redshirts are transformed into large Dungeons and Dragons dice. Mr. Spock uses his telepathic powers to learn that the Kelvans have assumed human form so that they will be able to function on the Enterprise; in reality, they are large creatures with one hundred tentacles, unsuitable for life aboard the spaceship. When it becomes apparent that the Kelvans are starting to develop human reactions, Kirk uses the aliens' newfound emotions against them. Eventually, the captain convinces Rojan that, by the time the Enterprise reaches Andromeda (a journey of 300 years), the aliens' descendants will be virtually human and considered inferior by their own race. Instead of returning to Andromeda, the Kelvans decide to colonise the uncharted planet where they were found.
Given how long it has already taken the Kelvans to reach the Milky Way, it seems to me that Rojan and Kelinda are very quick to shirk their responsibilities: their ancestors dedicated their lives to travelling beyond their own galaxy, but Rojan is willing to give up all plans of conquering new worlds just because he likes how it feels to be human. It's a weak and rather unconvincing conclusion to the episode, but one that I am willing to forgive because of Barbara Bouchet as Kelinda, who has to be one of the sexiest women of Star Trek. She looks absolutely amazing throughout, so it's no wonder that Kirk finds time to add her to his long list of women that he has snogged (who cares if she's really an immense creature with numerous appendages?).