Reagan Had Nothing To Do With It... Jimmy Was The MAN!!!
Iran released the embassy hostages because of Carter's negotiations, not in spite of them
The boring and emotionally unsatisfying truth is that the Carter administration secured the Americans' release through protracted negotiations — and by releasing millions of dollars to the Iranian government.
A little background is probably in order here. On November 4, 1979, a militant student group called the Muslim Student Followers of the Imam's Line stormed the US Embassy in Tehran, occupied the grounds, and took the more than 60 American diplomats and embassy staff there prisoner.
The hostages were held for 444 days. During that period, the Carter administration tried to secure their release through a military operation, which failed catastrophically, and then through a series of secret negotiations.
The negotiations were protracted and very messy, largely because of tensions between Iran's hard-liners and more moderate factions within their government. On several occasions, the Carter administration believed it had reached a final agreement, only to see the deal scuttled at the last minute by Iranian leader Ayatollah Ruhollah Khomeini.
But Carter and his negotiators kept working through the very end of his presidency, and eventually, at the last possible moment, they succeeded. On January 19, 1981, the US and Iran signed the Algiers Accords, an agreement brokered by the Algerian government that secured the hostages' release in exchange for concessions by the US, including sanctions relief, the release of frozen Iranian assets, and the creation of the Iran–United States Claims Tribunal that would remove cases against Iran from US courts.
The hostages were released the following day, January 20, 1981 — the day Reagan was inaugurated.
https://www.vox.com/2016/1/25/10826056/reagan-iran-hostage-negotiation