Agamemnon has a hot temper and and a prideful streak. His arrogance toward the people around him angered the gods. He also angered them directly by saying things that were disrespectful to the gods. He is self-absorbed and although he reconciles with Achilles in Book 19, he shirks personal responsibility blaming Fate, Ruin, and the gods.
Artemis becomes angry at Agamemnon because he says that Artemis herself could not have done it better after Agamemnon shoots a deer. Agamemnon's vanity causes him to disrespect Artemis, and he must pay for it.
Agamemnon also angers the gods in his conflict with Achilles, the son of a goddess. His mother has Zeus promise to teach Agamemnon a lesson because he treated Achilles so badly. Zeus sent Agamemnon false dreams, and the Greeks suffered great losses. Agamemnon's pride and arrogance causes the Greek army to suffer because he refuses to allow Achilles to have more than he does.
Apollo's anger toward Agamemnon was caused when the latter demoralized the priest of Apollo by refusing to return his daughter.
Agamemnon is amoral and self-serving, so he's unable to give the gods the respect he is supposed to. He likes himself too much, and at times, considers himself to be god-like, which is totally unacceptable to the gods.