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The negotiations that led to the accord had begun in 1968 and had been subject to various lengthy delays. The major breakthrough came on October 8, 1972. North Vietnam had been disappointed by the results of the its latest offensive, and feared increased isolation from major communist powers. Nixon's efforts at détente significantly improved U.S relations with the Soviet Union and the People's Republic of China, who had been backing the North Vietnamese military effort. The North Vietnamese significantly modified their position and became willing to allow the South Vietnamese government to remain in power. The South Vietnamese were not happy with the terms of the draft agreement proposed, and for a time it looked as if the agreement would go off the rails. But the U.S. brought great diplomatic pressure upon the South Vietnamese to sign the peace treaty even if the concessions they wanted could not be achieved. Nixon pledged continued substantial aid to South Vietnam.
Nixon ordered the Operation Linebacker II bombings of North Vietnam in December of 1972 to demonstrate that he was still committed to supporting the South Vietnamese. These operations were also designed to keep North Vietnam at the negotiating table. Nixon threatened that South Vietnam would be abandoned it did not agree to the peace proposal and the South had had little choice but to agree.
On 15 January 1973, Nixon announced a suspension of offensive actions against North Vietnam. Henry Kissinger and Le Duc Tho of North Vietnam met on January 23rd and signed off on a treaty that was basically identical to the draft of three months earlier. The agreement was signed by the leaders of the official delegations on January 27th. Kissinger and Tho were later awarded the 1973 Nobel Peace Prize, but Tho refused to accept it.
Below is a YouTube video of Nixon making the announcement, in which he used the phrase "peace with honor."