What brings about the atonement?
Ramban: The live bird that carries with the owner's sins. 1
Like the Sa'ir ha'Mishtale'ach on Yom Kipur (Ramban). He does not require an Asham and a Chatas - like he does by Nig'ei Adam (Refer also to 14:18:1:1), because the punishment of Nig'ei Batim is not as direct as that of Nig'ei Adam. Consequently, the Kaparah of the birds, the cedar-stick (the crimson thread) and the hyssop will suffice.
What happens to the dead bird and the live bird?
Refer to 14:7:2:1-2 and the notes there.