Why does Homer refer to Agamemnon's story in book eleven of "The Odyssey," starting at line 436?
Homer brings up the story of Agamemnon while Odysseus is visiting Hades as a way to add a bit of foreshadowing to the story, or a way to build up some suspense.
First, remember that Agamemnon was one of the mack-daddies of the ancient world, comparable to Odysseus himself. Yet, look what becomes of even this great hero:
"As soon as he had tasted the blood he knew me, and weeping bitterly stretched out his arms towards me to embrace me; but he had no strength nor substance any more, and I too wept and pitied him as I beheld him."
He's just another shade among shadows. Without that little taste of blood, he wouldn't have even been able to recognize Odysseus. Partly, seeing Agamemnon is a way to humble Odysseus and show him the fate that awaits even great heroes.
More than that, though, is the substance of what Agamemnon has to say about his demise. His wife and her evil boyfriend Aegisthus have set a trap for him:
"He asked me to his house, feasted me, and then butchered me most miserably as though I were a fat beast in a slaughter house, while all around me my comrades were slain like sheep or pigs for the wedding breakfast, or picnic, or gorgeous banquet of some great nobleman."
Notice how this is some foreshadowing of what is going to come for Odysseus upon his return to Ithaca: a feast, a fight, and a slaughter. Agamemnon is quick to add that he doesn't think that Odysseus' wife will attempt the same thing, but implores the hero to:
"'Be sure, therefore,' continued Agamemnon, 'and not be too friendly even with your own wife. Do not tell her all that you know perfectly wellyourself. Tell her a part only, and keep your own counsel about the rest. "
In essence, he warns Odysseus not to completely trust his wife upon his return. More importantly, he gives Odysseus this piece of advice:
"...do not tell people when you are bringing your ship to Ithaca, but steal a march upon them, for after all this there is no trusting women."
Indeed, this is exactly what he does, taking a disguise and scoping out the situation at home before revealing himself to the suitors or his wife.
So, to sum it up, Agamemnon's story is included here to show the fate that even the greatest heroes have waiting for them, as well as to put into Odysseus' head the fact that his wife might not be trustworthy after all these years. Lastly, he suggests a course of stealth when approaching his future home.