Ramban writes about this in his commentary to Vayikra 16:8.
Rather than translate the whole piece, I will link to this summary and analysis by R. Ezra Bick.
Some key quotes:
The Ramban begins by explaining that azazel refers to some sort of evil power (as opposed to the explanation of Rashi that it means a "hard and rocky place"). He first cites the Ibn Ezra, who hints that the explanation of azazel is found at the end of thirty-three. The Ramban, who, as we know, believes in not disclosing secrets, nevertheless here feels free to explain the Ibn Ezra, since, as he states, Chazal have already done so in several places. The reference of the Ibn Ezra is, as R. Chavel explains in a footnote, to the thirty-third verse following the present one, which is "and they shall no longer offer their sacrifices to the demons…" (17,7). The Ramban then cites Bereishit Rabba which associates the sa'ir hamishtalei'ach (the "sent goat") with Eisav ("ish sa'ir" – a hairy man). The Ramban, of course, understands this not as the historical figure of the brother of Yaacov, but as the twin and opposite figure to Yisrael, the power of evil. He makes this explicit in his next quote, from Pirkei d'Rabi Eliezer, that identifies the destination of the se'ir hamishtalei'ach as Samael. We now have only to understand who is Samael.
The quote from Pirkei D'Rabi Eliezer is as follows:
That is why on Yom Kippur they would give Samael a bribe to not cancel their sacrifice, as is written, "one lot to God and one lot to azazel," the lot of God is a burnt-offering, and the lot of azazel is a goat of sin-offering, and all the sins of Israel are on it. Samael sees that there is no sin in them on Yom Kippur. He says to God: Master of the worlds, you have one people on earth who are like the ministering angels in heaven – just as the ministering angels are barefoot, so Israel is barefoot on Yom Kippur; just as the ministering angels neither eat nor drink, so Israel does not eat or drink on Yom Kippur, just as the ministering angels cannot bend, so Israel stands all Yom Kippur; just as the ministering angels, peace serves as an intermediary between them, so Israel, peace serves as an intermediary between them on Yom Kippur; just as the ministering angels are free of all sin, so Israel is free of all sin on Yom Kippur.
God hears the testimony of Israel from their accuser and He atones for the altar, and for the Temple, and for the priests, and for all the congregation.
It is clear from this quote that azazel is Samael, who is the "accuser" of Israel; in other words, the satan. The Ramban goes on to explain.
See, they have told his name and his actions. And this is the secret of the matter. For they used to worship other gods, who are the angels, offering them sacrifices which are for them a sweet savor (rei'ach nicho'ach)…. Now the Torah totally prohibited the acceptance of their divinity or any worship of them, but God commanded that on Yom Kippur we send a goat to the desert to the prince who rules in desolate places, which is appropriate as he is the master of (that place), and from the emanation of his strength comes destruction and desolation, for he is the cause of the stars of the sword and blood and wars and quarrels and wounds and plagues and division and destruction, and, in general, the soul of the sphere of Mars. And his portion among the nations is Eisav, who is the nation who inherits the sword and wars. And among the animals, (his portion is) the goat, and in his portion are also the demons who are called mazikim in the language of the Rabbis, and se'irim in the language of Scripture, for both he and his nation are called se'ir (= goat, demon, and another name for Edom, the land of Eisav).
The intention of the se'ir hamishtalei'ach is not that it be an offering from us to him, God forbid, but rather that our intention should be that we are fulfilling the will of our creator who has commanded so. The parable for this is if one were to make a banquet for the master, and the master would command the one making the banquet: "Give a portion to this particular servant of mine." In this case, the one who makes the banquet is not giving anything to the servant, and is not honoring him, but rather everything is given to the master, and the master is giving a reward to the servant. And he has fulfilled his master's command and has done, in his master's honor, all that he was commanded. But the master, out of concern for the giver of the banquet, wished that all his servants take part in it, so that they praise him and not belittle him.
What does the service of Yom Kippur express? The Ramban gives a parable of a feast, to which the servant is invited so that he not be left out. I think that the meaning of this is that the service of Yom Kippur works by bringing about a unity of the different powers, the different servants, as it were, of God. On Yom Kippur, the Jews are like the angels, as the midrash in the Pirkei d'Rabi Eliezer makes clear. We make a banquet in God's honor, and it is important that all God's servants be included, because by partaking in the banquet they all become part of the unity of Israel's service of God. We don't serve the principle of evil, God forbid; we show that we understand that even the principle of evil is subject to God and dependent on Him. That is the opposite of serving evil; it is making, or rather demonstrating, that evil is subservient to God.