How did the U.S. and other countries negotiate taking Iranian banks off the SWIFT network?

They did not negotiate per se, it was the result of European Union's and US sanctions against Iranian banks associated with Iran's nuclear program, including the Central Bank of Iran.

Iran sanctions

In January 2012, the advocacy group United Against Nuclear Iran (UANI) implemented a campaign calling on SWIFT to end all relations with Iran's banking system, including theCentral Bank of Iran. UANI asserted that Iran's membership in SWIFT violated U.S. and EU financial sanctions against Iran as well as SWIFT's own corporate rules.[22]

Consequently, in February 2012, the U.S. Senate Banking Committee unanimously approved sanctions against SWIFT aimed at pressuring the Belgian financial telecommunications network to terminate its ties with blacklisted Iranian banks. Expelling Iranian banks from SWIFT would potentially deny Iran access to billions of dollars in revenue and spending using SWIFT but not from using IVTS. Mark Wallace, president of UANI, praised the Senate Banking Committee.[23]

Initially SWIFT denied it was acting illegally,[23] but now says "it is working with U.S. and European governments to address their concerns that its financial services are being used by Iran to avoid sanctions and conduct illicit business."[24] Targeted banks would be — amongst others — Saderat Bank of Iran, Bank Mellat, Post Bank of Iran and Sepah Bank.[25] On 17 March 2012, following agreement two days earlier between all 27 member states of the Council of the European Union and the Council's subsequent ruling, SWIFT disconnected all Iranian banks from its international network that had been identified as institutions in breach of current EU sanctions and warned that even more Iranian financial institutions could be disconnected from the network.

Source: Society for Worldwide Interbank Financial Telecommunication

Also read the press release that SWIFT itself posted on the issue.

SWIFT instructed to disconnect sanctioned Iranian banks following EU Council decision