The setup for "Man Trap" is straightforward; the Enterprise has stopped at planet M-113 to administer required yearly checkups to Dr. Robert Crater and his wife Nancy. There is a small added complication in that Nancy Crater is an old flame of Dr. McCoy's. Kirk, McCoy, and
And then things become quite eerie; Dr. McCoy, clearly still attracted to her, mentions how she hasn't aged a day, and indeed, she looks very young compared to McCoy, in her 20s perhaps. We see Kirk grinning at this, and McCoy introduces her... and we see that, to Kirk, Mrs. Crater appears to be quite a lot older, with little touches of gray in her hair. Ensign Darnell appears to be unable to take his eyes off her... which is not surprising, as we see that to him, Nancy Crater appears to be a barely-20 sultry-looking blonde bombshell ("... looks just like someone I left back on Wrigley's Pleasure Planet..."). Nancy leaves to go find her husband and bring him back; Ensign Darnell, who had been sent outside after his staring and verbal faux pas, sees the blonde Nancy walk out, give him unmistakable "come hither" signals, and walks off; Darnell, after a moment, follows her. Robert Crater arrives sometime later, apparently having missed Nancy along the way, and tries to convince everyone to leave -- after resupplying them, especially with salt -- and not perform these intrusive examinations.
Trek fans will of course be familiar with the episode; subsequently Darnell turns up dead, and so do other crewmen, until it is revealed that "Nancy Crater" died a year or two ago, replaced by an alien creature, native to M-113, which has the ability to change shape (apparently through psychic illusions). Dr. Crater had established a relationship with it originally due to not wanting to kill it because it was the last of its kind, using salt -- a vital nutrient apparently rare on M-113 -- to gain its trust; the nature of the later relationship is implied, and is not something they could have gotten into in the 60s!
Overall, it's a well-paced episode with excellent tension building. By modern standards it has of course a number of flaws -- the use of female characters is minimal, though not entirely token, and this will tend to stand out -- but as a first introduction to the series it was an excellent introduction to the main points of the series: multiracial, male-and-female crew on a military vessel, the three main characters including an apparently nearly emotionless alien, on a mission to explore and to solve unusual problems that they might encounter. This includes a fascination with the alien and an unwillingness to simply destroy anything that might be dangerous.
The "salt vampire" is one of the classic images from Original Trek, and with good reason; given the monetary and technical limitations, it is a terrifying creature, with a face that somehow includes sadness/pathos and hungry monstrosity into a single very alien visage. It might look somewhat silly today, although I still find the head/face creepy, but for its time it was exceedingly well done.
All in all, a very good first episode.