Palamon and Arcite, the noble cousins—though they also call themselves brothers—have been thrown into prison by King Theseus. The cousins are languishing in a tower that overlooks a garden in which Theseus's beautiful sister-in-law Emily can often be seen taking a walk.
The very moment that Palamon and Arcite lay eyes on this vision of pulchritude, they immediately fall in love with her. But in actual fact, this makes their lives just a little more unpleasant. Imprisonment is bad enough, but to be stuck in prison while suffering from extreme lovesickness is even worse.
When Palamon hears that Arcite feels for Emily the same way as he does, he can't believe his own ears. At first, he thinks that Arcite might be joking, but Palamon's cousin is deadly serious in proclaiming that without Emily's mercy and grace, he is as good as dead.
Now that he knows that Arcite's on the level, Palamon is furious. As far as he's concerned, Emily is his. Why? Because he fell in love with her first. And once he did, he told his woes to Arcite, who was supposed to be his loyal and faithful confidant. We can understand, then, why Palamon is so indignant that Arcite should also have expressed his undying love for Emily. It seems like he's going against the knightly code and betraying his cousin.