Richard Carpenter got the idea for the song when he watched an old Bing Cosby movie. The movie was about a songwriter, and throughout the movie, they would keep referring to the greatest song that this writer had ever written (called "Goodbye to Love", of course), but the movie's audience never actually got to hear this song. Richard thought "Goodbye to Love" was a great title for a song, and he set about composing it! With his creative genius inspired, the words and the tune flowed forth naturally.
The other interesting aspect was Tony Peluso's bravura guitar performance. He had to fill in a spot between Karen's lines, and what began as a bit of bold improvisation turned out to have started a cultural trend in pop music. Pop songs thereafter began to feature "power ballads" with "raging guitar solos", as he put it in an interview. Peluso was charmingly modest about his role when he said he took "a tiny bit of credit for being there and playing it", but he gave all credit for the idea to Richard Carpenter, calling him "the guy with the great ideas".
This clip has those interviews:
As the latest (and hopefully not final) piece of "Goodbye to Love" magic that I have had the pleasure to experience, I came across this work by the famous graffiti artist Banksy: