The conflict between Achilles and Agamemnon over the sex slave Briseis may seem somewhat trivial on the face of it. But to both these men with big egos, this is a matter of honor.
As far as Achilles is concerned, Briseis is his; she belongs to him like any war booty. Disturbing as it is to us today, it was quite common for warriors in ancient civilizations to take women as spoils of war and use them as slaves or concubines. Achilles is one of many such warriors who believes himself entitled to such a glittering “prize.”
So when Agamemnon appropriates Briseis for himself, Achilles is absolutely furious. He feels he's been disrespected and dishonored as a warrior, even though as king, Agamemnon is perfectly entitled to take whatever he wants from one of his subordinates. Even though Agamemnon has a perfectly valid reason for snatching Briseis from Achilles—he has to in order to replace his own slave whom he had to give up to appease the wrath of the gods—Achilles is so full of pride that he cannot accept what he regards as a personal affront to his dignity as a warrior.
Of course, Briseis's dignity doesn't enter into the equation at all. As both a woman and a slave, she doesn't really count in the hyper-masculine society depicted in the Iliad.