What are Odysseus' reasons for slaying the suitors in The Odyssey?
Odysseus kills the suitors at the end of The Odyssey because they take advantage of his family and his property. Their disrespectful behavior is something he cannot forgive when he arrives home.
Penelope's suitors spend years trying to convince her to marry them. They are a boorish lot who drink and eat everything that Odysseus's lands produce. They are not kind men; Penelope clearly tells them that she does not want to marry, and they continue to pressure her and insist she must give up Odysseus for dead and choose one of them.
When Odysseus returns home and wins the contest for Penelope's hand, he reveals himself to the suitors. He shoots Antinous through the throat; the other suitors try to bargain for their lives, but he refuses to let them leave. He even kills the priest, who he says likely prayed for Odysseus to never make it home.
In the end, everyone who Odysseus considered complicit in the bad treatment of his wife and property is killed. Odysseus, his son, his men, and Athena participate in the battle.
The final bloody showdown at the end of Homer's The Odyssey consists of Odysseus ruthlessly murdering Penelope's suitors. He shows them no mercy, while also having Athene on his side to aid him in murdering all of them. Athene also deters the angry mob of the suitors' relatives after the massacre. With everything back to normal, Odysseus is reunited with his family. But why would he relentlessly murder all of the suitors?
The suitors are taking all that is Odysseus's. They are living on his land (and making a real mess), eating all his livestock, treating his servants poorly, and moving in on his wife. Also, Odysseus has spent years on this treacherous journey home only to find these men taking advantage of all that is his. He has no remorse and little reason to offer them mercy, not to mention the Gods are on his side as well.
In my opinion, Odysseus kills all the suitors because they are trying to take away what is rightfully his. This would be a great concern to a person today, but to a person from Odysseus's time and place, it would be even more of a big deal. To him, the actions of the suitors would have been an affront to his honor.
The suitors are trying to take away his home and his wife. They try to kill his son. They are trying to destroy the things that would be most important to a Greek warrior/aristocrat of the day.
They had been trying to court his lady, they had eaten his livestock and taken advantage of his property. Even though the suitors were ready to pay him back and make retribution for what they had taken, Odysseus wouldn't hear of it.
Plus, Tiresias had commanded him to kill all the suitors, go a ways inland and build a monument to Lord Poseidon.
So I think you could say two of his reasons were to avenge them for their behavior and to be obedient to the gods.