As a child of Pluto who has returned from the dead, you understand the veil between worlds better than most. You can control the Mist.
-- Hecate (IV paragraph 3)

Importance: Hazel learns that she has an important role to play when they reach the House of Hades. She needs to realize that she is more powerful and important than she believes she is. The fate of her friends and possibly the world rests on her defeating the witch in the House of Hades. She is the only one who can do it.

He is the void that consumes all magic, the cold that consumes all fire, the silence that consumes all speech.
-- Gaea (IX paragraph 2)

Importance: Leo learns that Clytius is a formidable foe and he will be waiting for the group when they get to the House of Hades. Leo's fire is useless against the giant, but the demi-gods will have to find a way to defeat him...