The Enterprise provides transport for Ambassador Alkar (Chip Lucia), who is on a mission to broker the peace between two warring factions on a distant (or nearby, I guess, if you are traveling on the Enterprise) world. He travels with an elderly woman that he claims is his mother; when Troi meets them in the transporter room, the mother immediately sees Troi as some kind of threat and warns her to stay away from Alkar. Soon after, the mother dies of unknown causes. Alkar asks Troi to perform a sacred ceremony with him, after which Troi starts behaving rather ... bizarrely.
"Man of the People" is a hodgepodge stew of tired Trek clichés, perhaps the most tired one being the perfunctory iteration on the "two warring factions to which the Enterprise is bringing an ambassador." These two cultures, never seen on screen, are completely arbitrary placeholders, giving us no plot to invest in.
The other cliché is the reliable "crew member behaves strangely because of unknown alien influence"; in this case Troi slowly begins to lose her mind and turn into some sort of oversexed crazy person. There is, to the episode's credit, a priceless scene where Riker shows up to Troi's quarters, sees what she is wearing, and starts smiling ... and then just as quickly stops smiling once he sees the random crewman in her bed. The scene at least knows that it's funny — as does the scene where Troi "counsels" a crewwoman and tells her, basically, to stop complaining about trivial crap or transfer off the ship.
The bizarre behavior is about the only real fun to be had here. Much of the rest is a slog and a bore, especially once Troi begins aging rapidly and the crew must then race against the clock to figure out what Alkar has done to her, and why, before she dies. Turns out Alkar is using Troi (as he used the woman before her, who was not his mother) as a "receptacle" to telepathically shunt off all the negative emotions that he experiences during his oh-so-taxing political negotiations. This is a pretty lame plot explanation, if you ask me. It's made no better by the fact that Alkar attempts to justify it with an ends-versus-means speech that's a completely obvious straw-man argument. (And when Picard immediately rejects said straw man, Worf then has his phaser conveniently pickpocketed.)
I also tire of countdown clocks (especially of the Sickbay Suspense™ variety) that are so difficult to buy into. The clever solution to the problem is to kill (yes, kill) Deanna long enough that Alkar will be forced to "release" his hold over her. She can then be revived, but only within 30 minutes. Inevitably, we have dialog informing us there's (gasp) only 1 minute and 45 seconds left to revive Deanna! Couldn't the Enterprise simply have warped really far away to break the connection? I guess Alkar can maintain this telepathic connection over light-years of space. Of course, once the telepathic connection is broken, Troi automatically de-ages and is perfectly fine. I should keep that in mind the next time I decide to drink a bottle of bourbon every night for the next 20 years. As long as I stop at year 21, I'll be fine!