why are agamemnon and achilles fighting?

Agamemnon and Achilles are fighting because Agamemnon has taken charge of Achilles' slave Briseis claiming her as his property. Achilles already hates Agamemnon, so this final act of disrespect, causes Achilles to withdraw from the war.

Agamemnon, the commander of the Greek forces, had ordered Achilles to hand over his slave girl Briseis after Agamemnon had been forced to give his own slave back.