Worf's first assumption is that this is some sort of challenge ritual, e.g., personal combat. Picard at first believes this as well, and throws away the knife that Captain Dathon throws to him, refusing to fight. Dathon replies, frustratedly, "Chaka...when the walls fell." Riker's first attempts to communicate with the Tamarians fail, and he orders Worf to assemble a team and take a shuttle down to save Picard, hoping the Tamarians won't push things that far. Picard, meanwhile, is still getting nowhere-Dathon appears friendly, and throws Picard a brand with which to light his own campfire, but communication is still seemingly impossible.
The shuttle tries to go down and is hit by Tamarian phasers-but only with enough force to make them go back. Riker is puzzled. Geordi thinks that given enough time, he might be able to punch through a very tight beam and get Picard off, but it'll take at least a day to get ready. Riker orders him to get on it, and orders Data and Troi to work on establishing a communication of some sort.
They find very little at first-"Darmok", used as a name of some sort, has 47 different meanings in nearby systems. After Troi expresses her frustration at the situation, they try again, this time with "Tanagra". It also has many meanings-but the meanings for a particular planet combine nicely ("Darmok" is a mythical hunter-hero, and "Tanagra" is a mythical island). Meanwhile, Picard examines Dathon's campsite when he finds Dathon gone, and discovers some sort of captain's log. But just then, Dathon hurries back, and frantically tries to give Picard a knife again, saying "Darmok and Jilad at Tanagra!" repeatedly. Picard refuses-but then a loud growl is heard from not very far away. "Darmok, and Jilad..." says Dathon with resignation, "at Tanagra."
Picard then takes the knife, realizing that the problem is not with Dathon. Since ship's sensors are picking up this creature (whatever it is) moving towards the two captains, Riker hurries Geordi along with the transporter, despite Geordi's warning that it might not work. Meanwhile, as the creature approaches, Picard makes an intuitive breakthrough, and finally realizes that the Tamarians communicate via metaphor, by citing examples from their own culture. "Sucat, his eyes uncovered!" exclaims Dathon in elation. However, the creature's attack easily breaks through the defenses of both men, and Dathon is sorely battered. Picard moves to help-and is promptly seized by a transporter beam. The beam isn't strong enough to get him, though, and Picard (who screamed out against the beam when it came) is returned to the surface, just in time to see the creature vanish, and Dathon fall unconscious.
As the situation worsens in orbit (the interference field has now been strengthened, Data and Troi conclude that although they know how the Tamarians communicate, they don't know what the examples mean, and Riker prepares to fire on the Tamarian ship to remove the field), Picard "talks" with a wounded Dathon. Eventually, he pieces together the puzzle-Dathon intended for the two of them to come down to the planet and fight a common enemy to form a bond between them, just as Darmok and Jilad did against the beast of Tanagra. Picard, in return, tells Dathon a tale of Gilgamesh and Enkidu- just before Dathon succumbs to his wounds and dies.
When Riker receives word that Dathon's life signs have vanished, and that the creature has been detected moving towards Picard again, he decides they're out of options, and they open fire. The field is removed, and Picard is saved just in time. Both ships trade shots several times, and the Enterprise is close to destruction, but Picard manages to communicate the facts of the situation enough to the Tamarian first officer that both sides leave peacefully; not necessarily as friends, but certainly not as enemies.