The Enterprise receives a distress call from an unexplored sector of the galaxy. It is traced to a planet named Scalos and Kirk leads a landing party down to the surface to investigate further. They can find no trace of life except, apparently, for some insects that they can hear buzzing but which remain unseen. One of the crewmen, Compton, disappears suddenly before their eyes for no apparent reason. Failing to find out what has happened to him the landing party beams back aboard to find, shortly afterwards, malfunctions starting to occur all over the ship.

A strange, alien device appears in life support and it seems that the buzzing of insects has followed them back to the Enterprise. Kirk orders a cup of coffee and, as he drinks it, the movements of the bridge crew appear to become slower and slower until they are 'frozen' in a stationary position. An alien woman appears who introduces herself as Deela. She is queen of the Scalosians and explains to Kirk that it is not his crew that have slowed down but he who has speeded up, matching the highly accelerated existence of the Scalosians.

To the bridge crew he just appeared to vanish in the same way that Compton had done. Compton is already aboard and has got used to the situation but he is fatally injured trying to protect Kirk when he attempts to gain access to the device in life support. Deela tells Kirk that the radiation levels on the planet accelerated their existence and has left all the male Scalosians sterile. She plans to put his crewmen into suspended animation until they are needed for reproduction.

Kirk has been chosen to be her partner and she intends to take him back down to the planet with her but he manages to buy himself some time by sabotaging the transporter. Meanwhile, Spock has been analysing the buzzing noise that turns out to be highly accelerated Scalosian conversation. McCoy analyses the remaining contents of Kirk's coffee and an alien substance is discovered for which an antidote is produced.

Spock drinks some of the Scalosian water and is accelerated, enabling him to help Kirk. Having completed repairs to the transporter, Deela prepares to beam down but Kirk manages to overpower her and steal her weapon. He and Spock make their way to life support where they destroy the alien device. Deela is allowed to return to the planet and Kirk takes McCoy's antidote, returning him to normal speed. Spock extends his stay in the accelerated condition for a while in order to be able to carry out repairs to the ship more quickly.