The two new songs, "Free As A Bird" and "Real Love", were originally John Lennon demos. The three surviving Beatles added instrumentation to the demos and released the songs as singles in conjunction with the release of the special. In addition, John hadn't filled in the middle eight section of the demo for "Free As A Bird", so the surviving Beatles wrote a new section for that, which, according to Paul, was one of the reasons for choosing the song. It allowed them some input.
The idea for this retrospective first came up in 1970. Apple Corps collected as much archive footage of the band as they could from around the world. Neil Aspinall then assembled a 90 minute documentary tentatively titled "The Long and Winding Road". Plans for the completion of the project lay dormant until 1980 when all four of the former members of the Beatles agreed to a television special which was to have featured a mini-concert at its conclusion. The project was to have been broadcast sometime in the early 80's. However these plans where interrupted when John Lennon was murdered on December 8, 1980.
The series was released simultaneously with an accompanying album and book. The three surviving Beatles had heard that original drummer Pete Best had fallen on hard times and in order for Best to get some royalties from the Anthology sales, they included early original recordings that were made with Best on drums, most noticeably the January 1st 1962 Decca demos. The amount of Royalties Best received haven't officially been made known, but based on sales figures, they are believed to be between £1million to £4 million.
George Martin, who had produced most of the Beatles' recordings, turned down an invitation to produce the posthumous "Free as a Bird" due to hearing problems (though he managed to produce and direct Anthology's soundtrack releases).
Long before the surviving Beatles collaborated on the unfinished John Lennon song "Real Love," the song in its original demo recording can actually be heard in the opening credits of the documentary "Imagine: John Lennon."