Odysseus, with the help of Telemachus, slays the suitors for two main reasons. First, they were disrespectful to him and his household, especially the first ones he and his son target with arrows and spear. Second, they were disloyal to Odysseus as ruler of Ithaca in their attempts to displace him by taking his wife.
As for disrespect, Antinous, for example, was the most impertinent of the suitors, and for this he is killed first by Odysseus, with an arrow fittingly put through his throat. Telemachus kills Eurymachus, who was disrespectful to him, with his spear. Ktesippus is killed by Odysseus for having thrown a cow's hoof at him, a deeply disrespectful act.
Athena, though she is in the background to help if needed, lets Odysseus and Telemachus fight off the swarm of suitors, who badly outnumber them, without her help. This reasserts their right to authority in the household and shows Odysseus in particular to be a strong warrior, worthy of command of his kingdom despite his long...
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