In The Odyssey, in what ways do we see Odysseus change due to his meeting with the sirens?
It actually is quite hard to note a particular change in the character of Odysseus that occurs due to his meeting with the sirens. It is clear that he is successful in being the only man ever to hear their song without succumbing to it thanks to his men, who tie him to the ship's mast and are given strict instructions to ignore him. Let us pay attention to what the sirens say to Odysseus and how they try to tempt him:
'Come here,' they sang, 'renowned Ulysses, honour to the Achaean name, and listen to our two voices. No one ever sailed past us without staying to hear the enchanting sweetness of our song--and he who listens will go on his way not only charmed, but wiser, for we know all the ills that the gods laid upon the Argives and Trojans before Troy, and can tell you everything that is going to happen over the whole world.'
What is interesting about this section of the epic is that straight after surviving this, Odysseus seems to act in a rather foolhardy way. In spite of having been told by Circe not to bother challenging Scylla, he nonetheless puts on his armour and thinks that he will be strong enough to face Scylla in battle. Perhaps we can link this to the grandiose notions that the song of the sirens fills him with. This, however, is not a cause and effect that is identified explicitly in the text.